.profile is only read when you log in interactively. When rsync connects to another machine to execute a command,
~/.profile are not read.
If your login shell is bash, then
~/.bashrc may be read (this is a quirk of bash —
~/.bashrc is read by non-login interactive shells, and in some circumstances by login non-interactive shells). This doesn't apply to all versions of bash though.
The easiest way to make rsync work is probably to pass the
--rsync-path option, e.g.
rsync --rsync-path=/home/elbarna/bin/rsync -avP /home/user/.profile hpux3:/home/user/.profile
If you log in over SSH with key-based authentication, you can set the
PATH environment variable via your
~/.ssh/authorized_keys. See sh startup files over ssh for explanations of how to arrange to load
.profile when logging in over SSH with a key.