rsync remote files over SSH to my local machine, using sudo privileges on local side, and my personal SSH key

Try this:

sudo rsync -e "sudo -u localuser ssh" -a myserver:/var/sites/example.net/ /var/sites/example.net/

This runs rsync as root, but the -e flag causes rsync to run ssh as your local user (using sudo -u localuser), so the ssh command has access to the necessary credentials. Rsync itself is still running as root, so it has the necessary filesystem permissions.

Just improving on top of larsks's response:

sudo rsync -e "sudo -u $USER ssh" ...

So in your case change rsync -a myserver:/var/sites/example.net/ /var/sites/example.net/ to sudo rsync -e "sudo -u $USER ssh" -a myserver:/var/sites/example.net/ /var/sites/example.net/.

With regards to @larsks' answer, If you have your key loaded into the ssh agent, which is my use case, you can instead do:

sudo rsync -e "env SSH_AUTH_SOCK=$SSH_AUTH_SOCK ssh" /source/path /destination/path

Instead of the double sudo.

My use case, if anyone is interested in replicating, is that I'm SSHing to a non-root sudo-er account on remote A, and need to rsync root-owned files between remote A and remote B. Authentication to both remotes is done using keys I have on my real local machine and I use -A to forward the ssh-agent authentication socket to remote A.