Terrestrial radar systems only cover so much area. The latest Terminal radar system the FAA uses, the ASR-11, has a working range of 60 nautical miles. The Enroute ARSR-4 has a range of 250 nautical miles. The trade-off is between update time and range; a radar antenna that spins faster can provide quicker updates, but only in a smaller radius.
It is very possible that a trans-oceanic flight is beyond primary and secondary radar coverage, and for this reason separation in oceanic regions is effected by employing non-radar or procedural control procedures: basically the aircraft use long-range HF radios to report over certain fixes at certain altitudes at certain times, and ATC spaces aircraft longitudinally (in-trail) or laterally (on different tracks) so that aircraft at the same altitude are not too close.
So if an aircraft is outside of radar coverage, their 7700 secondary radar beacon code will not be picked up until they get close to land. Once they do, alarms will go off in the control room, the tag will flash on the scope, etc. This will not necessarily get passed to third-party sites, though it could be; I know, for example, that FlightAware gets a data feed from the FAA with information about the position and altitude of civilian flights (as long as they haven't submitted a request to block their tail number). Whether squawk code is part of this data, I don't know.
(As Dave points out, transponders that do not have ADS-B capability do not continually broadcast their set beacon code—they wait until they get a signal requesting that they do so. This signal could come from an ATC secondary radar antenna or a nearby TCAS-equipped aircraft. If neither are within range, the beacon code will not be broadcast in the first place.)
But also: FlightAware and other websites are fed by ADS-B ground stations which pick up on the transponder signals sent by aircraft (and ATC systems are using their own standalone ADS-B sites not associated with primary radar sites). These sites are, of course, also not sitting in the middle of the ocean. But the signals are not encrypted; there is nothing to prevent someone from writing a script to send a tweet every time an emergency code is detected, and indeed people have done just that.
The next big thing is space-based ADS-B, where the ADS-B receiver station is located on a satellite. This would allow quasi-radar control to be effected in oceanic regions. Of course this doesn't help the amateur planespotter unless they can get a Raspberry Pi sent to space.