I guess it really depends on your instrument, taste, style and the group you are playing with. While studying Cello I actually had a lot of lessons with string quartet where we were analysing the score (harmony) for intonation and it happens quite often in modulations that enharmonic equivalents were used to distinguish whether a chord belongs to the old or the new harmony. And sometimes we really needed to make a difference between an e flat and a d sharp to match an open string or to get a desired tension.

For me the enharmonic equivalent is usually just a totally different harmony, so that is what I tune to. As a result they are quite different notes. I try to do that consciously - also while playing with fixed pitch instruments when possible (like the grandparent comment explained).