If two players execute key commands nearly simultaneously on a Piece, it can lead to extremely undesirable and unreliable behavior. I first noticed this because both players deleted an eliminated piece - it would double report it in the chat window. I’ve had players report some “impossible” values for Global Properties, meaning no Piece activity could result in those GP values, yet they kept occurring. I finally decided to run a test by manipulating the same Piece over-and-over simultaneously on my laptop and desktop. Low and behold, this sporadically led to module-breaking GP values. I tested on Vassal engine 3.4.13 and 3.6.6 and the behavior is the same.
It’s likely always been this way.
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That would have been my guess. I’d also guess this is a Vassal 4 thing, not a Vassal 3.7 fix, but I figured there is no harm in reporting.
Yes, it’s a thing we’re designing for in V4 from the start. It would not be straightforward to fix in V3 at all.
In the meantime, the best approach is to wait your turn. 
In general, people know to wait their turn and not to touch other players’ pieces. The trouble arises when players attack neutral powers. The likelihood of simultaneous clicks is MUCH higher with neutral units because no player owns them. I’ll just sternly warn them with Alerts…