The honest answer is that it depends heavily on the severity of the condition and how it manifests day to day.
First, some context worth having. Mental illness in the prison population is extraordinarily common. A significant portion of people who end up incarcerated are dealing with some form of untreated or under-treated mental health issue. That is not an excuse for the choices that led them there, but it is a reality of the environment. You are rarely the only person in the unit carrying something.
For inmates whose condition is manageable and does not visibly affect their behavior, the experience is closer to normal. They navigate daily life, stay out of trouble, and their diagnosis is largely their own business.
Where it gets harder is when the condition affects behavior in ways that are visible to other inmates. Prison culture does not have a lot of patience for erratic, unpredictable, or socially disruptive behavior, regardless of the cause. Bullying happens, and people who seem vulnerable or different can become targets. That is a real risk and it would be dishonest to minimize it.
When someone genuinely cannot function safely in the general population, the facility will typically move to separate them. Psychiatric units, protective housing, or dedicated mental health wings exist at many larger facilities for exactly this reason. The goal from the institution's perspective is managing liability and maintaining order, which in practice means isolating people whose condition creates problems rather than treating the underlying issue with any real depth.
If someone you care about is going in with a diagnosed condition, make sure it is documented and disclosed at intake. That paper trail matters.