On one machine I have two Windows accounts with two separate dropbox accounts. I installed Greenshot on one account and authenticated the Greenshot installation with that account. I then switched users to the other account and saved a screenshot to Dropbox. This saved and was created in the Dropbox account of the original account.
I understand why this happens as the installation of Greenshot is the same for all users, but Dropbox has a separate installation for each user. So I think that the second Windows/Dropbox account should have to re-authenticate.
Interestingly I had to re-authenticate using the same Dropbox account on different computers.