DEVONthink stores all data in a dtBase2 file. It's a container which contains the file and other settings. I have a DB called CSE where I keep my web archives, bookmarks and I want to use it from my work and personal accounts in the computer. I created the DB under the personal account. So it belongs to that user. It is advised to close the DB before opening by other user. Since the work user does not have permissions, it opens in read-only mode. To fix this we can add read-write permission to the DB file. Now if we try to open it will show "Database seems to be already in use" warning and when clicked on continue anyway it won't open. The solution as suggested by cgrunenberg is to quit DEVONthink, remove the .dtBase2 extension, which will now appear as a folder. Now add the work user with read-write permission for the folder and apply changes for enclosing files and folders. Add back the .dtBase2 extension. Open the DB and it will work just fine!
NB: It's adviced to never use the open the same database by mutliple users at the same time.