Tail a file from ssh and mirror to a local file

Yes, you can usee tee for that:

ssh  -t remotebox "tail -f /var/log/remote.log" | tee -a /var/log/local.log

This way output will be printed on both stdout and copied over into /var/log/local.log on the system you are running ssh command from.

Just redirect the stdout.

ssh -t remotebox tail -f /var/log/remote.log > local.log

To append to local.log:

ssh -t remotebox tail -f /var/log/remote.log >> local.log

This will write to the local server.

The remote server would only be written to if you include the redirect in the quotes:

ssh -t removebox "tail -f /var/log/remote.log >> remote.log"