Rinnegan occurs so rarely that the Sage of Six Paths was considered a myth, so those multiple Rinnegan's are surely not distinct ones.
Nagato's bodies seemingly having their own Rinnegan is an ability provided by the Six Paths of Pain technique. According to Narutopedia:
The Six Paths of Pain (ペイン六道, Pein Rikudō) is an Outer Path technique
devised by Nagato after being crippled by Hanzō and rendered emaciated
from the numerous chakra receivers embedded in his back by the Demonic
Statue of the Outer Path. By piercing corpses with chakra receivers, a
wielder of the Rinnegan can manipulate the bodies as though they are
The article further says:
Because of the fact that he was crippled, Nagato channelled the
abilities of the Six Paths into six different bodies (one ability per
body). The only common features shared by these bodies were the
Rinnegan they gained from this technique, the chakra receivers each
were pierced with and their bright orange hair, which had been dyed to
match the Deva Path's natural hair colour.
In other words, the multiple Rinnegan's are like "projections" or copies of Nagato's own Rinnegan, transmitted through the chakra rods. This theory can be further confirmed based on the following:
In Tobi's version of the Six Paths of Pain technique, his six bodies received a Rinnegan and a Sharingan, and he had a Rinnegan and a Sharigan at the time.