You could do a Rest API call to Jira/Confluence /rest/api/3/myself on the server by generating a signed JWT yourself using the sharedSecret you received in the post install (if you use ACE, this is done automatically for you, otherwise you will need to calculate qsh and add the calling user accountId in the sub which you can get from the JWT passed in the query string).
Probably best to store the value either with your own user object in the database or in an httpOnly cookie to avoid having to call the API for each request.