SQL availability groups log maintenance
When you run the log backups from a secondary server, when the log backup is complete the secondary replica will tell the primary replica that the log backup has been completed and what the last LSN to be backed up was.  The primary replica will then mark the VLFs which were backed up as inactive.  That change (which is logged) will then be transferred to the secondary replicas.

In other words, no you don't need to take backups from the primary replica as well as the secondary.