I don't have any experience with wal-e, but based on what I learned during my research (went with barman), there is no need to to install barman on the primary or the standby server, just a separate instance with proper user/permissions.

This separate instance is then called from primary's "archive_command", which usually looks like: archive_command = 'rsync -a %p barman@10.0.0.1:/var/lib/barman/main-db-server/incoming/%f' and once rsync'd barman does it's own magic on it (thru cron). Based on my observations, wal-e is more powerful if you want to store archives remotely (and is primarily an archiving solution), and barman is kinda the official backup solution (2ndQuadrant is a big contributor in the community) plus it offers point-in-time recovery.

Barman + Repmgr (which can do an automatic failover or a manual switchover) is a powerful combo. If only there were reliable tools to determine the master and slave (like Redis Sentinels)