Back on topic...
Despite what the movies say detecting another sub that doesn't want to be detected is not easy. Especially if you don't really believe that it's out there.
This surfaced (ahem) a while ago when one country's submarine, in a wargame situation, was able to photograph the propellers of the opposing side's aircraft carrier.
This from inside a windowless tin can.

One might assume France & England have developed incredibly effective stealth submarines. Then again, maybe their ability to detect submarines really sucks.:)

Really an event that had to happen sometime. Two submarines with sound absorbing skin, moving slowly, passive sonar only simply won't see each other.

I know a guy who served on a boomer -- very dull was his summary -- but their strategy is straight-forward. Steam out from Newport News (or wherever) to the Gulf Stream, run at the slowest speed at which attitude, depth, and direction control are easily maintained, and basically drift with the Gulf stream until they're nearly to Ireland. At that point, another sub enters the stream off the US and the first one heads home.

This strategy can hardly be unique to the USN. I'll bet they all do it. So we're not talking about two subs meeting at random in the North Atlantic, we're talking about two meeting in the Gulf Stream which near the US is only 100 to 200 km wide +/- 40 km.

Had to happen sooner or later.

This is similar to how aircraft vectors work. Airplanes generally travel through invisible 'roads' in the sky marked out by map/radar. I can't find the article, but I recall a few years ago a civilian 757 almost collided with a stealth F-117 bomber (yes, it is technically a bomber even though it has a F designation).

Offtopic: I just found out that all of the F-117s were taken out of service in 2008 and been replaced with F/A-22s!

There's a beacon system of air lanes marked out on every air chart and domestic flights are directed to these and passed along from controller to controller. A military jet would normally be visible to controllers and military pilots know where (their maps show) where these routes are. When I was a kid in the RCAF we were supposed to stay the hell out of them.