Typically private keys are not shared. You have a different key for every use, which typically includes every device. I'm not sure that's the right decision, but it's the typical one.
However, most private key storage technologies support encryption. They often say "password protection" but it's implemented as strong encryption. So in my opinion, as long as you "protect" with a suitably impossible-to-guess password, then the security is largely under control. I wouldn't put it up on pastebin simply because there's no need to invite unnecessary risk.
As an example, I have an SSH key that was used as the only point of access for a set of servers. That key was protected by a random password and was never stored unencrypted, ever. That encrypted key, along with other important files, was saved to a folder that was mounted using encfs, controlled by another random password. The backing-store for the encfs filesystem was a folder synchronized between workstations by Dropbox.
So to get at my key, you'd have to get in to my private Dropbox folder (non-trivial, but not impossible), guess my encfs password (significantly more difficult), and then guess my private key password.
For one person to get lucky enough to hit all three is unlikely enough for me to feel pretty confident in the syncing arrangement. And I get all the relevant important secret documents synced between all the associated machines across Linux, OSX, and Windows, and it just works. Each "secret" folder is only encfs-mounted on-demand when they're needed, so even though the files are technically present everywhere that Dropbox syncs (good for backup purposes), they're not available without the encfs password.