Video
There are interior Canon XF305 (and now XF-705 as of 2021) camcorders all throughout the USOS portion of Station, but the crew could manually turn them off, unplug them, place the lens cap on them, or face them at the wall if they so desired and Mission Control could do nothing about it.
Generally, the crew leaves the cameras plugged in and the physical power switch on at all times. The cameras are powered off by ground-based flight controllers at the end of each work day and only powered back on starting at morning DPC (Daily Planning Conference). Mission Control generally asks permission to "come aboard" prior to activating the cameras.
During the workday, there are six simultaneous video downlinks used for "over the shoulder" situational awareness for flight controllers or science teams in order to assist the crew members in their planned activities. The "six pack" video is available whenever there's Ku-band coverage.
Voice
There's no audio automatically downlinked from the ISS. Crew members have a PTT handset that's wired into the ATU (Audio Terminal Unit) panel located in each module. They can talk to Mission Control with the blue handset/microphone shown below whenever there's S-band or Ku-band coverage.
(image source)
If a crew member is performing a task in a glovebox or during EVA, they can wear a VOX headset which transmits whenever a certain noise threshold is exceeded. That way it's not constantly broadcasting white noise, and when they need to speak they don't have to free up a hand for the PTT.

Privatization
Any SG voice loop and video downlink can be privatized so only a small subset of "need to know" personnel can hear/view. This is done for any medical experiment procedures a crew member is performing, or for sensitive research that may be proprietary. One example that always gets voice/video privatized is Rodent Research performed on the ISS. There are special subsets of each flight control discipline that handles those procedures.