Do we have a special bin folder for a particular user in Mac OS X?

In Linux we have /home/<user>/bin to put special binaries that only that user will execute

This is not standard Linux. Under Linux, or Unix-like systems such as OS X, you can have a directory for executables anywhere you want, as long as:

• it is in your $PATH
• the partition it's on allows execution of files (see the noexec option for mounting file systems)

Simply create /Users/your-username/bin and add it to PATH – you're done. See: What are PATH and other environment variables, and how can I set or use them?

You can also create /usr/local/bin, then run sudo chown -R /usr/local to have it owned by your account. This directory is not used by default in OS X, and the Filesystem Hierarchy Standard recommends it as a place to put software installed by the administrator. If you're the only admin user of that machine, that works pretty well.