How do I find out why my computer hangs during logout?

I use multiple user accounts on my Mac (For security purposes, make sensitive/private information physically inaccessible in certain contexts). If I start and shutdown the computer in a normal single-user flow, everything works fine. But if I am logged in with both user accounts at the same time, i.e. I switch account without logging out first, then I am unable to log out from any account. I can switch fine between each logged in account; but logging out, and therefore also shutdown/restart doesn't work. The screen goes black, and the computer just hangs. No keyboard or mouse interaction have any observable effect. I am left with a forced shutdown by holding the power button. I have not yet tried making an SSH connection from a third computer; which I guess could help identify the cause of the problem; but I don't know what to look for either. I assume that it's some deadlock in a program that is running once for each user account, but how do I determine which?