Let $R$ be a ring and let $M$ be a right module over $R$. We say that $M$ is faithfully flat as a right module if the functor $R \otimes_R -$ from left $R$-modules to abelian groups that preserves and reflects exact sequences. Faithful flatness for a left $R$-module is defined analogously.

What is an example of an $R$-bimodule that is faithfully flat as a right module, but not faithfully flat as a left module? Or what is an example of an $R$-bimodule that is faithfully flat as a left module, but not faithfully flat as a right module?