Typically when someone is placed on suicide watch they would be placed in a specifically designated and designed observation cell. The cell would be devoid of most things one would be able to hang themselves from or harm themselves with and would provide a good view for an officer or a camera to maintain 24/7 observation. The person in the cell would have their belongings taken away save for a tear-resistant smock of some kind, and the list goes on. The only way to be removed from such a watch would be at the direction of the Sheriff/Warden and the Chief Psychologist.
This is in a perfect world with perfect staffing and with enough observation cells for severely suicidal offenders. In the real world, things get a lot weirder and tragically so. You may have a severely limited number of designated observation cells; my Level III, 1100 bed facility has one designated observation cell for mental health holds. The other cells that can be used in a pinch have blind spots. You're supposed to conduct 15 minute checks in my department. People can be lazy, and it isn't unlikely that these checks can be penciled in, especially by those working long OT shifts. Suicidal persons get really creative; you can kill yourself with even a small amount of elastic from the inside of your underwear in about 10-12 minutes. Anecdotally speaking, there's a story from a member of the Blue Check Twitteratti about an inmate committing suicide 20 feet from a CO in an observation cell. If the CO had adjusted their chair about three feet or stood up and walked over, they would have seen what was happening and may have been able to prevent someone's death.
Of course, recent events seem to indicate that the person in question was removed from suicide watch prior to their successful attempt. This was probably done because of the OT being incurred and the danger of not staffing "more" critical posts.