Note: The prefix can be important to avoid mistaking messages from the remote system as messages from the local system.
That said, there is no way to turn off the prefix, but they are all written to stderr. You could redirect/capture/filter the stderr of git push to do what you want.
A rough way of doing might be something like this:
git push ... 2>&1 | sed -e 's/^remote: //'
It sends stdout to a pipe and makes stderr goto the same place. At the other end of the pipe, sed reads the combined output and deletes any remote:
prefixes. This should be okay since it we are unlikely to see remote:
prefixes in the stdout stream. Combining stdout and stderr like this is generally acceptable for interactive use (since they were probably going to the same tty device anyway), but it may not be a good idea for use in automated contexts (e.g scripts).