I played a guy in doubles that would routinely drop shot off of our serves
As always there are no hard-rules. Depending on level of play of all 4 guys on the court, and agility/mobility some things do work and some others don't. So just because pro-s do it in a specific way may not mean it would work for everyone at every skill level. Mostly at pro levels, the server should be able to get to it, and should be able to finish it or at least avoid getting hammered an overhead of next shot and the SP closes the angles, leading to all 4 players at net. But it is also true that it will be extremely hard to do a drop-shot on a return, at pro-levels.
But now if we bring the agility level down, what maybe happening is that the server is barely getting to the drop shot, possibly popping it up for an easy overhead for the opponet net guy, who then hammers it down through the middle of the court. Now if the servers partner (SP) attempts to get there (considering the agility and anticipation), he also would be late to this ball, mostly ending up in an awkward position trying to hit the ball moving away from him, leading to a weak shot DTL or middle, which the opponent net guy can angle it away to the opposite short corner (since now that side of the court is pretty open, assuming the server did not have the agility to move quickly forward and sideways and cover it). The real issue is that either of the move calls for good agility on the guy who is not hitting the ball (partner), and your teams skill level may not be to that par. So the opponent team is able to take control of the point, without much agility from their side.
So first thing is to understand that this is a "defensive situation" for your team (even though it may not be for higher levels). Which means, the most effective method is to avoid getting into that situation. How? Try Australian formation (Server and Server partner at the same side of the court). Especially if this drop shot is on the adserve. This immediately takes away the short angle dropper, and now the drop has to be straigh ahead (DTL), which won't be going away from the court, and hopefully now the server or SP has a better chance to reach there with more comfort. It is also possible that the returner may not even be able to do that comfortably. Also even though the duece side of the court is now appears open, by inviting the returner to hit to there, you are actually getting him to play his weaker shot (back hand down the line) to your servers stronger side (forehand down the line), which immediately puts your team in a stronger position, and the rest is about the SP intercepting and helping to finish and bring error from opponent.
Now if this is happening on duece side, you may have to re-analyze how opening up the left side of the court works for your team in a few points. If the opponent has a good DTL forehand, and server has a weaker DTL backhand, and the opponent net guy is assertive, it may not work good on duece side. But you may be able to change the pattern immediately from DTL by throwing off a lob or short angle.... again.... depending on skills and comforts on shots.
Now, even if you end up in this situation, the first and foremost is to accept that you are now in DEFENSE (or you both have to work on your agility). So think of the basics, when you are defense. Cover the easiest targets and buy time if you can (lob it cross court straight to the baseline guy for example), and see if you can somehow get back in the point.