I thought the initial speculation centered on a Navy P-3 Orion aircraft in the area armed with practice missiles that had no warheads. Someone even posted a picture of a P-3 on the ground with its missile complement after Kallstrom or someone connected to the investigation denied that P-3's were armed at all. The theory was that a practice missile from the P-3 missed its drone target and went on to hit TWA 800 in the center of mass, ripping it apart and neatly explaining all of the odd symptoms that investigators later found. No one said that a Navy ship was involved, which would make the number of Navy personnel needing to be silenced in the event of such a mishap quite a bit smaller.
The one thing that makes this all difficult to sort out is not just the inherent inaccuracies in any eyewitness account, but the clear disconnect between different events as described by individual witnesses (mainly a function of the distances involved).
For example, a witness who claimed that they saw a flash and then almost immediately heard an explosion will understandably assume there is some connection between the two discrete events. But in the case of TWA Flight 800, this is not the case at all. Sound travels much slower than light, and since Flight 800 was about nine miles offshore when it went down, there would be a 45 second lapse between when an explosion occurred and when a person on the shore would have heard it. So it is entirely possible that a person saw a flash and heard an explosion almost simultaneously, but the sound could not have been caused by the flash in question. Investigators would have to work backwards to a point 45 seconds earlier to determine what exactly caused the sound that was reported to have been heard on the shore.
Just think about the time intervals we're talking about here. 45 seconds is an incredibly long period of time in any kind of incident like this.