In order to use Facebook Messenger, you must be a user of Facebook. I believe that the only way that you could use Facebook Messenger without having an account of your own would be to use someone else’s Facebook account and then access their Facebook Messenger. However, you have to be on Facebook in order to use their messenger.
There are other messengers to use other than Facebook, but sometimes you have to have an account with them too. It would be very difficult if not impossible to use the messenger without not being on Facebook. I don’t think you could access the messenger without being on Facebook and to be on Facebook you have to have an account. Maybe in the future it will be possible to use Facebook messenger on your own.