This is EXACTLY what I do.  The labels that will peel off (retail
pharmacy) I stick on a piece of paper and shred, and the ones that
won't (my mail-in pharmacy) I use a black permanent marker to mark out
all print.  I do the same thing with everything that has my name,
address or any other information on it before it goes in my trash.
I've read about too many folks out there that went through legal and
financial difficulties from ID theft, to take any chances.

When you're on stuff like Oxycodone, Vicodin, Duragesic and other
things like that, especially if your neighbors and friends know you
have a chronic illness and are in pain a lot, it only takes a word to
the wrong person to cause problems.  I don't want to put temptation in
anyone's path, or give them an opportunity to cause problems for me.
The government already makes it difficult enough for chronic pain
patients to receive the stronger medications, and puts restrictions on
the doctors in writing them.  Why contribute to the problem.  All
these steps may seem silly and unnecessary, but this isn't aspirin
we're talking about.  I don't want my name on a prescription bottle
that some druggie has on them if they get arrested somewhere for
something.

This is just my perspective.  I worked retail pharmacy from 1974
through 1995 off and on, until the year I was paralyzed from Guillaine
Barre Syndrome. I finally let my pharmacy certification expire 2 years
ago when I realized I would probably never be able to work again.  But
over the years, I called the cops for quite a few prescription
forgeries.  With the high volume pharmacies around now, and everything
computerized, the criminals are finding new and improved ways of
stealing drugs.
Mag