We should bear in mind that the AMRAAM missile doesn't necessarily need the SARH guidance to come from the launch aircraft though.
We've seen plenty of "hand off" missile shots over the years. I remember the RAF boasting about the new capability to do just that from the Typhoon and F/A-18A/B's I'm aware have long had that capability as well.
F-35 will most assuredly have that capability and I'd be astonished if modern AEW&C and AWACS aircraft can't provide it too...
Well, the Cooperative Engagement Capability gives ships the ability to fire on targets observed by E-2Ds, F-35B/Cs, and F/A-18C/D/E/Fs, as well as allowing the aforementioned aircraft to fire on targets detected by each other, so the Navy certainly has that capability.
What Spudman said is correct, though -- during the terminal phase of the missile's flight, something will have to emit a radar signal. Either the missile will have to use ARH homing, or the aircraft around the missile will have to provide fire control to allow SARH guidance. Until we manage to put AESA radars on AAMs (which doesn't seem to be happening soon, due to the huge amount of power electronically scanned arrays use), those emissions will be easily detectable by the targeted aircraft. The question is preventing the location of the missile from being known, and thus preventing an electronic attack or evasive maneuver from being successful. ARH homing would give away the missile's position, but keep the F-35 hidden. SARH, on the other hand, would hide the missile, but not the F-35. IMHO, it would seem like SARH homing would be best in this situation, as it would "hide" the missile, and the F-35 would only be visible for a very short period of time -- most likely a tenth of a minute or less -- and only its bearing would be known, not its range. Of course, I may be wrong about that, and obviously the situation the F-35s find themselves in will dictate which is the best option (which is probably why the AMRAAM includes both SARH and ARH modes -- to provide the pilot with options to deal with unpredictable situations).
At that point, it just comes down to when the missile would be detected by the targeted aircraft's MAWS. If the MAWS is radar-based (i.e. Russian systems), the range at which the missile would be detected would probably be quite short, due to both the low-RCS of the AMRAAM and the low power of the MAWS radar. However, I obviously don't have any hard figures to suggest exactly what range the missiles would be detected at, so the question remains -- at what range would the missiles be detected by the targeted aircraft, how much time would remain between detection and impact, and would that time frame be long enough to permit an electronic attack or evasive maneuvers?
