Joerg
I tried it both. I changed the user and group of the files in the /var/lib/grommunio-web/sqlite-index folder to groweb:groweb and set the permissions of the index.sqlite3 file to 0660.
After did so, the websearch was woking but I got this failures when I was reindexing:
[ERROR] Failed to insert message 10/1918698: attempt to write a readonly database
[ERROR] Failed to insert message 10/1918699: SQLite query failed: 8
I think, the way with the wrong permissions is right, but this is in my case not the solution.