I can't help you with Android, and I'm not really using FB Messenger on my iDevices. However, afaik the only method to logout from the app itself, without needing another app, is to remove the app, and reinstall it in case you need it again.
No I m asking for apple device. Before this apps updating no issue because it's open all the time if anyone else of ur family member use iPad if he click messenger for his own use he can also view yr then switch to his account.
JA I check in help section read no logout option that's why I uninstall FACBOOK MESSENGER app because it's a lick of privacy not out side but between the family members, if this feature only for iPhone then it's ok but iPad use I don't feel it's a good idea.
It's still possible to switch the account. Tap "Me" at the bottom of the app on your iPad, then you'll find the option to change the user.
If you wish to completely sign out of FB Messenger, you'll have to do so in the FB app. In More - Settings - Account Settings - Security - Where you're logged in, you'll find apps and devices connected to your FB account. Look for the messenger app, and tap the x at the right.