Each user on each system needs to have a private ssh key, and a public ssh key.
No, they don't. Private keys belong on your closest system, on an encrypted volume. Often, you will only need one.
If the OP uses ssh to go from system1:user1 to system2:user2,
and then wants to use ssh to go from system2:user2 to system3:user3, are
you saying that only system1:user requires a public key, and that
system2:user2 can ssh out without having *any* public key?