Copying Linux users and passwords to a new server

Solution 1:

Start with

/etc/passwd - user account information less the encrypted passwords 
/etc/shadow - contains encrypted passwords 
/etc/group - user group information 
/etc/gshadow - - group encrypted passwords

Be sure to ensure that the permissions on the files are correct too

Solution 2:

I did this with Gentoo Linux already and copied:

• /etc/passwd
• /etc/shadow
• /etc/group
• /etc/gshadow

that's it.

If the files on the other machine have different owner IDs, you might change them to the ones on /etc/group and /etc/passwd and then you have the effective permissions restored.

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Solution 3:

Be careful that you don't delete or renumber system accounts when copying over the files mentioned in the other answers. System services don't usually have fixed user ids, and if you've installed the packages in a different order to the original machine (which is very likely if it was long-lived), then they'll end up in a different order. I tend to copy those files to somewhere like /root/saved-from-old-system and hand-edit them in order to just copy the non-system accounts. (There's probably a tool for this, but I don't tend to copy systems like this often enough to warrant investigating one.)