Well, it's not possible to sign out from within the Messenger however there is an easy workaround to do so if you're using an Android gadget. iPhone, iPad and WP users can likewise logout without uninstalling the app but that needs utilizing Facebook's web user interface, either from your mobile browser or desktop. Though, this isn't an easy to use way but will do the needed job.
How To Sign Out Facebook Messenger
Go to phone Settings > Applications and open the Messenger app. Click the 'Clear information' option and now open Messenger. You'll now able to log into Messenger utilizing a different Facebook account. Note: If you've Facebook app installed and you're singed into it then Messenger will still reveal your formerly connected account that can be set up right away.
To check in utilizing a different ID, just click the 'Switch Account' alternative. In case you wish to unlink your earlier account from Messenger, just logout from Facebook app.
Note: Cleaning information will eliminate the app settings and all your messenger information (stored locally) such as photos. However, the messages and images will be synced again with your account. Follow the below alternate procedure instead, if you don't want the information to be wiped off.
Logout from Facebook App/ Messenger App on iOS, Android, Windows Phone and Desktop (Recommended Approach).
This is a very useful way to remotely logout from Facebook from any or all the gadgets to which your Facebook account is linked with.
It comes genuine helpful in case you loose access to a specific device and want to prevent unauthorised access to your account. The benefit of this workaround is that it does not wipes your regional Messenger data and deals with all significant mobile OS like iOS (iPhone, iPad, iPod touch), Android and Windows Phone.
To do so, log into your Facebook account from your desktop internet browser or use Chrome web browser to log into facebook.com on your mobile phone and go with desktop version of the website. In Facebook, browse to Settings > Security and click "Where You're Visited" choice.
Here you can see all your Facebook active sessions along with details like Gadget type. Select the Messenger one and click "End activity". Now open Messenger on your phone and it'll immediately log off (await a short while and ensure you're connected to Web.).
Dear Facebook, please do not annoy users by managing basic functionality. Take that!