We have this Hybrid setup where few pilot users are migrated to office365. Now the problem started, wherein on-prem users who have access to Calendars for the users who are migrated are getting random password prompts. Does anyone come across this situation and a solution?
Did this issue persist via OWA?
Folders can be accessed cross forest in many scenarios, but they are not fully supported by Microsoft as outlined in Permissions in Exchange hybrid deployments.
I suggest you remove the permission and re-assign it. It is better to create a new profile for test.
The best solution is to move both mailboxes to Exchange Online.