If I use Rise from the Grave to return a creature from my opponents graveyard to my battlefield and they have a Prized Amalgam in their graveyard, does it still trigger for them and come under their control?
Indeed it does. This is actually a really interesting interaction.
Prized Amalgam looks for a creature to enter the battlefield that’s either cast from your graveyard or entering the battlefield from your graveyard. It doesn’t care under whose control the creature is entering the battlefield, just that it’s doing so and that it came from your graveyard.
This means that Prized Amalgam will trigger when your opponent reanimates another creature from your graveyard with Rise from the Grave or any other replacement effect. (It would also trigger for weird stuff involving Gather Specimens.)