Hypnotic Siren says "You control enchanted creature." This is a continuous effect that's present as long as Hypnotic Siren is on the battlefield enchanting a creature. Once it's gone, that effect goes away and you'll lose control of the creature, giving it back to whoever had it before.

Since you used Hypnotic Siren as just one example, though, I should mention that not everything works that way.

If a card simply reads "Gain control of X" and doesn't mention a duration (e.g. "until end of turn"), that effect lasts indefinitely. For example, Beguiler of Wills reads:

{T}: Gain control of target creature with power less than or equal to the number of creatures you control.

That effect lasts indefinitely; you'll get to keep the creature for the rest of the game, unless someone else has a way to gain control of it.

Hypnotic Siren says "You control enchanted creature." This is a continuous effect that's present as long as Hypnotic Siren is on the battlefield enchanting a creature. Once it's gone, that effect goes away and you'll lose control of the creature, giving it back to whoever had it before. (In common situations, anyway - see this question for details.)

Since you used Hypnotic Siren as just one example, though, I should mention that not everything works that way.

If a card simply reads "Gain control of X" and doesn't mention a duration (e.g. "until end of turn"), that effect lasts indefinitely. For example, Beguiler of Wills reads:

{T}: Gain control of target creature with power less than or equal to the number of creatures you control.

That effect lasts indefinitely; you'll get to keep the creature for the rest of the game, unless someone else has a way to gain control of it.