So what if you are at the grocery store or post office and you make eye contact with an ASP you know. Do you go up, hug her, and talk. Look the other way? I am curious to hear from the Ladies and gents.

I would nod to her and then leave it up to her if she wanted to hug, talk, or shake hands. I definitely wouldn't make a big scene.
jhb

I believe that it's expected that client & provider be discrete at all times...  that includes the grocery store!  Many clients would freak at the very thought of a strange woman talking to them...  with the risk of needing to explain to friends & companions just who that was.  Even if YOU"RE ok with that, there's the risk of outting her!  That she might be forced to explain to others just how you know each other.  NEITHER of these is cool.
Exceptions might be if both of you are single & she's previously accompanied you in public, say to lunch or a drink.  It may also be that any such public date occured in a safe location & the grocery store might not be.  
The safest thing is to perhaps make slight eye contact, a little nod of the head & keep on about your business.  
skb

I would give a warm smile and hello and allow her to decide whether to nod back, say hello, have a chat, or what.

The morning after I had seen a lady, (Oddly enough, we were in the same hotel) we ran into each other in the hotel restaurant. I just glanced at her as I walked to my table. She was with another lady and a gentleman. As she was walking to the restroom, she seemed to go out of her way to walk past my table and gave me a big smile. We left it at that.

Something very similar to this happened to me at Walmart. I saw a provider that I had just seen a week ago, we really seemed to connect and she told me I was special. I went up to her and pinched her ass and gave her a wet kiss.  To my surprise she smacked me in the face and than some punk ass teenage kid told me I was an asshole. I think it might have been her kid.  How was I supposed to know.

Seriously, the general rule is to ignore each other.  At most a polite smile, just like you would give to anyone else you pass, but do not know.

Do not do anything that signifies that you know each other.   As an example, if you saw a woman that you had seen before, and you were with your wife, would you want her to come over and hug you?

A smile is a quite adequate response.   If the other person does want to be approached, that person can look away so as to say that they do not want to be acknowledged.

In fact if a woman I have seen were to do more than a smile if she saw me outside of a session, I would be rather angry because it violated the unwritten rule about what happens in the bedroom stays in the bedroom.

So just smile.  Even strangers smile at each other in public.

Maybe touching near the left eye with your left index finger to say: "I (eye) know you."?

That's just based on shooting from the hip.

Anyone else care to offer one?

If you should happen to be wearing a baseball cap, this would present itself an excellent opportunity to practice 3rd base coaching signals.  Something like this:

Right hand to left elbow
Left index finger to left ear
Right index finger to right eye
Left index finger to lips
Right hand massages left pectoral region
Left hand massages right pectoral region
Right hand... quick slap on the glute
Left hand makes hovering sign over crotch
Right index and middle finger to mouth
Deep inhale
While maintaining right hand pose, move right hand 8-12 inches away from mouth
Deep exhale

This should not draw much attention as all baseball fans know this is the signal for batter to bunt and advance the runner on first base.

That's the signal for the squeeze play, man on third base trying to score.

most of us are always trying to score. lol

when she was buying a car (A Bentley. Trust me, she's still VERY well known even though she doesn't do film anymore!)
She said that she was always amused at the supermarket by the guys who recognized her and were basically staring at her from around the corner at the end of the aisle. None would approach because they were usually with wifey!