When is it ok to stop signing off in a message thread?

Seems like people stop signing off much quicker with Facebook messages (btw if you have my email address I would much much much much much prefer you to email me please. Why choose to use the crappier messaging system?). Sometimes FB messages don't get a sign off at all.

When do you stop signing off? I'd probably only stop if the thread become 'document attached' or 'see you then'. Even then, a lot of people would go for the initial, right?