When RIP v2 authentication is enabled on 1 router it means it'll authenticate the updates coming from peer and not the updates being sent to peer.
Example here enabling authentication only on R2
Example here enabling authentication only on R2
R2#sh run int s0/0
Building configuration...
Current configuration : 193 bytes
!
interface Serial0/0
ip address 192.168.12.2 255.255.255.0
ip rip authentication mode md5
ip rip authentication key-chain cisco
Results of debug
*Mar 1 00:10:26.535: RIP: ignored v2 packet from 192.168.12.1 (invalid authentication)
But if we check routing table on R1 and R2
R1#sh ip route 2.2.2.2
Routing entry for 2.2.2.2/32
Known via "rip", distance 120, metric 1
Redistributing via rip
Last update from 192.168.12.2 on Serial0/0, 00:00:01 ago
Routing Descriptor Blocks:
* 192.168.12.2, from 192.168.12.2, 00:00:01 ago, via Serial0/0
Route metric is 1, traffic share count is 1
Router 1 still learns about routes from R2
But router 2 doesn't learn any routes from R1
R2#sh ip route 1.1.1.1
Routing entry for 1.1.1.1/32
Known via "rip", distance 120, metric 3
Redistributing via rip
Last update from 192.168.24.4 on Serial0/1, 00:00:21 ago
Routing Descriptor Blocks:
* 192.168.24.4, from 192.168.24.4, 00:00:21 ago, via Serial0/1 <<<<<<< From R4
Route metric is 3, traffic share count is 1