1,000ft. This was changed several years ago. As mentioned earlier, since you don't have anything to judge the distance when flying, they always seem closer than they really are.

Both planes will be informed of the other by ATC, they both then make a visual of the other aircraft and report back to ATC that they do in fact see the other plane. If there was an issue, the TCAS system would have warned the pilots that another plane was within the 1,000ft range and you would have felt some type of maneuver.

When I often worked the JFK-SFO, and would be in the cockpit during their physiological breaks, we often flew over and/or under other aircraft all the time. It's a lot more frequent than you realize.

I can't remember which flight it is, but from LAX-JFK, our 767 would often depart before Qantas 747. At some point over Nevada, they would ask to go around us since the 767 goes slower than the 747, and you would often see them fly-by 'close'. Pretty cool if you witness it!!!