I agree with David regarding the dialogue that takes place during the playing of classical music. When things are going well, there are ideas bouncing around all over the place, and when you're playing with people that are both receptive to what's going on and also contribute to the mix, then it's the same feeling as when you're at a session and everyone is in that groove. Then there are other times when you are not really "playing together" as much as you are just "playing near each other." But it's also different if you're playing in a small chamber group or in a large orchestra. You still have to listen and participate in an orchestra but you also have to play well with others , if you will.
Since New England Conservatory has appeared a couple of times in this thread, I thought I'd add a story. When I was at NEC we would sometimes get together on Friday nights for "Drinking Quartets." Two would go get beer and the other two would go get armfuls of music from the library. We would read and drink and ... well ... drink more. Did wonders for our sight-reading. It sounds funny but we found the best way to get better at sight-reading was to practice it. The quartets exapanded to sextets, octets, and then we invited other friends that played wind and brass instruments. It was a blast. Admittedly we had a rather high concentration of musicians per sqare inch there on Huntington Ave, but it looks like that link to ACMP would help you get a group together.
And please, no viola jokes.