It would depend on which two people are involved, but for myself, I would have to do a lot of thinking about why that feeling of "connection" exists and where it's coming from, psychologically. Sometimes a client that you think you might have a connection with is just an especially nice person compared to the rest. Or you think that there is something more there until you try to take your interaction out of the client/provider context.
I would keep it businesslike as long as no one's feelings were hurt.
Ladies/Gentlemen,
What do you do, if anything, when it's obvious that you have a connection when you see someone?
Do you just let that be when you see them?
Do you say something?
Do you shut off to keep it provider/client?
Just curious.
After all, we all have some connection hopefully or the session would be quite a bore.
Now, if you are talking about being head over heels for each other, then that can be a problem if it screws things up for you two.
For starters, it's best just to let it be, and don't mess it up with plans of eloping off and running away from your respective families. Just see each other and enjoy the ravishing moments you have.
These attractions are usually fueled by hormones, and within months or weeks the initial rush is over and then you can feel normal with the person.
If, after that time, you still have incredible hots for each other, that's when you have to take a hard look at what your life is about and decided what to do about it.
There's not right or wrong, but we all end up doing something about it.
Your hormones are involved; how sure are you that the other party feels the same or similar connection? Apparently you don't know or you wouldn't be asking here...
Give it time and if you still feel the same way after multiple encounters, then have a conversation AFTER you decide on your exit strategy... Because the answer you are about to get may not be the one you want, in more ways than one...
Breast of luck 
I have been seeing a provider that I absolutley adore.. she has been doing little or no volume and I have been helping her out as I can with issues that come up..and frankly, I feel that I am playing with fire.. She is very experienced.. ranked very high nationally in the top 100.. and has simply stepped back and has been reflecting on her future.. I am not allowing myself to fall for her (but it would be very easy to do so...) and you might think the same way..I plan on enjoying myself with her as much as I can..and she still is charging me and that's what I want.. keeps the lines drawn..
I saw her this week and she asked if I wanted another shot on goal.. I said "No".. FRankly I was enjoying post coital snuggling and talking to much.. I felt that working toward another might disturb that.. I have resolved myself to see some new girls now.. as I have to step away.. a little.. but can't wait until I see her again.. she pushes all of my buttons and I remind myself why she has practically all 10's.. she is good..and a wonderful person besides.. so, it's a dangerous game we play..
I would have to say we are all humans first...so yes, most definitely connections come when you least expect them, even in this world. And since things happen a lil differently in this world (the 'chicken comes before the egg' typically), I would say that 'time' is the true test for anything. And with time comes truth. And IMO if it's meant to be, it will be, naturally without effort. The only question that could arise is if you are both on the same playing field when that 'time' does arrive.
I wish you luck.
What do you do, if anything, when it's obvious that you have a connection when you see someone?
Do you just let that be when you see them?
Do you say something?
Do you shut off to keep it provider/client?
Just curious.
I would stop seeing the person probably.The two tend to not mix and match well.
It would depend on which two people are involved, but for myself, I would have to do a lot of thinking about why that feeling of "connection" exists and where it's coming from, psychologically. Sometimes a client that you think you might have a connection with is just an especially nice person compared to the rest. Or you think that there is something more there until you try to take your interaction out of the client/provider context.
I would keep it businesslike as long as no one's feelings were hurt.
If so, I shall hazard a guess and say that she has done a wonderful job of creating the GFE for you. While there are some real connections this is more than likely a business for her and she is giving you what you paid for, underline the paid for. I am not being a DB I am reminding you what this is from the get go.
All the smiles and eye gazing and other girlfriendly type stuff is done to make you feel like it is real. Take away the money and see what happens. Or tell her how you feel and see what happens, or see it for what it most likely is and keep on enjoying her just as you would any other person you paid for a service.