I'm in germany where the general voting must be secret, i.e. a vote that is given openly is invalid and must be destroyed. The voter can then get a 2nd sheet and fill it in in the cabin. (There seems to be some leniency in practice wrt. incorrectly folded sheets, where one can see the cross as the sheet is put into the ballot box, and there has been a scandal years ago when the Hessian parliament had a vote in secret mode [most of their votes are done openly] but some members took mobile phone shots of their ballot).
We do have a historical comparison when election law changed from optionally secret to mandatory secret: Not a perfect comparison since a whole lot of other things besides the mode of election changed, but: Volkskammer elections in the GDR were not fully secret: it was not mandatory to use the election cabin (and doing so was noted and had repercussions afterwards) Also, no cross anywhere meant "yes" to the unified list. So it was easy to show publicly and for others to confirm agreement. Any "action" on the ballot (= easy to detect) meant therefore that the one who did so did not say "yes" to the existing system.
Rules changed for the very last one: election cabin use became mandatory making the election fully secret plus a number of further changes.
In the 1986 elections (old rules), the unified list was approved with 99.74% yes votes. In the (last) 1990 elections, the PDS (successor party of the previous state party SED) got 16,4% of the votes.
Of course, not all of this drop can be attributed only to the new full secrecy of the elections: the 99+% needed a combination of lack of secrecy and actual power of the SED (ability to actually repress those who would not clearly vote as desired) and also "yes" being the default option etc. And that power was largely lost by the time the 1990 elections came up.
But with fully secret elections that power cannot be abused in this way.
Update wrt @gerrit's comment: yes, GDR elections were rigged in other ways as well.
Here's a secondary source I found that cites Lindner, Bernd: Die demokratische Revolution in der DDR 1989/90 about the communal elections on May 7th, 1989. These were still done the old rules, but in hindsight first signs of what in the end lead to the reunification can be detected. In summary, there was proof of rigging of the results of up to 10 percent more disagreement than the official 1.15 %.
I provide some more context, since it shows how "nudging" by making it, errr, unpleasant for people to excercise their existing right to monitor the vote counting helped with the rigging of the results. IMHO this is very much in parallel to the on-topic question how voting secrecy which is only a right but not mandatory was basically non-existent in practice.
(my emphases)
Vor allem aber fanden sich am 7. Mai 1989 im ganzen Land Gruppen von Menschen zusammen, die entschlossen waren, von ihrem gesetzlich verbürgten Recht der Wahlkontrolle Gebrauch zu machen. Sie wollten nach Schließung der Wahllokale an der öffentlichen Auszählung der Stimmen teilnehmen. Es war ein offenes Geheimnis in der DDR, dass die bei Wahlen üblichen Ergebnisse (fast hundertprozentige Beteiligung der Wähler und nahezu vollständige Zustimmung zu den Kandidatenlisten der Nationalen Front) nicht auf einer reellen Basis beruhten. […]
Niemand hatte bis 1989 gewagt, der SED- Führung und den ihr unterstellten Staatsorganen direkt nachzuweisen, dass sie Wahlergebnisse vorsätzlich fälschten. Gehörte dazu doch mehr als der entschlossene Mut eines Einzelnen. Ein wirklicher Nachweis über Wahlfälschungen konnte nur erbracht werden, wenn in nahezu allen Wahllokalen eines Ortes oder Wahlbezirkes neutrale Personen an der Auszählung der Stimmen beteiligt waren, um ihre Erkenntnisse anschließend zusammenzutragen und sie mit den in der regionalen Presse veröffentlichten offiziellen Wahlergebnissen vergleichen können.
Das setzte gemeinsame Absprachen und einen hohen Organisationsgrad voraus - alles "Tatbestände", die in der DDR zu einer Kriminalisierung wegen "antisozialistischer Gruppenbildung" voll ausreichten. Um so erstaunlicher die Zahl der Kommunen und Wahllokale, in denen sich am Abend des 7.Mai 1989 Bürger einfanden, um die Auszählung der Stimmen zu überwachen. […]
Die Ergebnisse der Überprüfung waren eindeutig. In nahezu allen Fällen konnte den Behörden Wahlfälschung nachgewiesen werden. Dabei hatte die SED-Führung schon klammheimlich "Zugeständnisse" an die Stimmung in der Bevölkerung gemacht: Das offiziell verkündete Endergebnis der Wahlen wies mit 1,15 Prozent den höchsten Anteil an Nein-Stimmen in der Geschichte der DDR auf. Blieben aber immer noch 98,85 Prozent Zustimmung für die Kandidaten der Nationalen Front (bei einer angeblichen Wahlbeteiligung von 98,78 Prozent). Dies widersprach deutlich den Ergebnissen der autonomen Wahlbeobachter. Die Differenz zu den offiziellen Angaben betrug in einigen Orten bis zehn Prozent.
Translation (by deepl translator and some editing from my side):
Most importantly, on May 7, 1989, groups of people gathered throughout the country determined to exercise their legally guaranteed right to monitor the election. They wanted to participate in the public counting of votes after the polling stations closed. It was an open secret in the GDR that the results usual in elections (almost one hundred percent voter participation and almost complete approval of the National Front's candidate lists [aka unified lists]) were not based on a real basis. [...] Until 1989, no one had dared to prove directly that the SED leadership and its subordinate state organs were deliberately falsifying election results. This required more than the determined courage of an individual. Real proof of electoral fraud could only be provided if neutral persons were involved in counting the votes in almost all polling stations in a town or electoral district, so that their findings could then be collated and compared with the official election results published in the regional press. This required joint agreements and a high degree of organization - all "facts"[German Tatbestände are legal facts, such as elements of a crime] that were fully sufficient in the GDR for being criminalized for "anti-socialist group formation." All the more astonishing was the number of municipalities and polling stations where citizens turned out on the evening of May 7, 1989, to supervise the counting of the votes. [...] The results of the monitoring were clear. In almost all cases, the authorities were proven to have falsified the elections. At the same time, the SED leadership had already made clandestine "concessions" to the mood of the population: the officially announced final result of the elections showed the highest percentage of "no" votes in the history of the GDR, 1.15 percent. However, there still remained 98.85 percent approval for the candidates of the National Front (with an alleged voter turnout of 98.78 percent). This clearly contradicted the results of the autonomous election observers. The difference from the official figures was as high as ten percent in some places.
Note that even if we could apply this rigging of actual votes guesstimate to the difference between the 1986 and 1990 elections, there are still other major differences besides manadory voting secrecy, from the whole situation to the fact that voters could for the first time choose among multiple options whom to vote for, and that those options proposed vastly different plans for the future.
It is well known that polls/predictions on elections can be quite off, and that highly contented elections are particularly difficult (remember Brexit, Trump 2016, in Germany we also had a few demoscopic predictions being widely off). For those predictions, people not openly admitting what they then vote (in secret) is one important known source of uncertainty, but again, not the only one.
More directly relevant data may be the differences between exit polls and actual election results. In particular for votes that go against the (perceived) public mainstream.
However, I only had a quick (and fruitess) search on this.
Not quantifyable, and only an anecdote. But for what it's worth, here's a personal experience:
I was member of a student association where management committee election rules specified secret election in a general assemby as the default mode - but also the additional possibility that a single candidate could also be elected by acclamation. Eventually, acclamation became in practice the usual mode of election. At some point there was a single candidate where several people would have liked to have a secret election because they thought the subgroup behind that candidate too powerful, but they did not dare to speak up openly. In the end, someone with "sufficiently neutral" standing heard, and requested secret voting in the assembly. IIRC, a more general discussion resulted, another candidate was found and the initial one did not get elected.
The "icing" on this story: this took place about 15 years after reunification, in "Eastern Germany". The association was quite proud of having historical roots close to GDR opposition and GDR citizens' rights movement (Bürgerrechtsbewegung), and many of the students came from such a background as well.
Nevertheless, it happened that we didn't realize how important secret voting is* until quite late.
* Even in not-so-important contexts, I mean, this was just some student club: the cost of speaking up with an uncomfortable request was minor (compared e.g. to the risk the GDR election monitoring people took). But it was nevertheless sufficient to keep several people from exercising a right they had and which they actually wished to exercise.
