Right after the weapon is dropped...
...you can use your free action to kick a non-heavy weapon up to 15 ft away (depending on terrain) -- at least at my table. Where the item ends up after the kick is determined by the DM.
You don't get to immediately take the weapon, but a benefit is that it doesn't create an opportunity attack upon you, which would happen if you bent down to pick it up. And the kick might place the weapon in reach of your ally to pick up later in the same round.
Admittedly this is a "house rule," but it doesn't run afoul of any RAW that I know of.
Page 190 of the PHB explains that you can "interact with one object or feature of the environment for free [i.e., without using an action]... For example, you could open a door during your move... or you could draw your weapon... The DM might require you to use an action for any of these activities when it needs special care or when it presents an unusual obstacle." (emphasis mine)
What makes this is a house rule is that (1) the PHB doesn't explicitly state that you can use your free object interaction to kick a dropped weapon, and (2) the part about considering whether such a kick "needs special care" could conceivably lead a DM to rule that the kick requires an action.
But my view is that if you did not already use up your one free object interaction per turn, and if you prevailed in the skills contest to disarm the weapon, then you can kick it.
In my experience, this has not created an imbalance in the game. It is used by the players only every once in a while. They have to spend a free object interaction to do it, and sometimes a foe picks up the weapon before an ally does. So it gives them an interesting option without automatically guaranteeing a better outcome.
Further details on the status as a "house rule"
Even though the PHB rule on free object interaction says you can "pick up a dropped axe" and "kick a small stone", that still doesn't answer all the questions about the situation.
As for "pick up a dropped axe", that does not mean you can avoid an attack of opportunity when picking it up, especially if it is close to a foe you are engaged with in melee. As for "kick a small stone", a weapon is considerably bigger and heavier than that. Therefore, a ruling by the GM that you can kick a just-now-dropped weapon without creating an opportunity attack, is a house rule, as I see it.
See also this question: How far away can dropped or disarmed weapons be kicked?