That's only half the story. To stay completely delta and gamma neutral, the market maker typically buys not just the share but additionally shorts a call with the same strike as the put.
When he shorts the call it means that someone else buys it. Probably another market maker, who might want to hedge as well, this time by selling the share and buying a put, etc..
When he shorts the call it means that someone else buys it. Probably another market maker, who might want to hedge as well, this time by selling the share and buying a put, etc..