When do you flip a coin for conditions
According to Rulebook, page 15:

Asleep
Turn the Pokémon counterclockwise to show that it is Asleep.
If a Pokémon is Asleep, it cannot attack or retreat. Between turns, flip a coin. If you flip heads, the Pokémon wakes up (turn the card right-side up), but if you flip tails, it stays Asleep.

Flipping for sleep occurs between turns. On the same page, there is a brief description of what between turns means:

Before the game continues to the next player, take care of Special Conditions in this order:

POISONED     2. BURNED     3. ASLEEP     4. PARALYZED

Then, apply the effects of any Abilities (or anything else that a card states must happen between turns). After both players have done these things, check to see if any affected Pokémon were Knocked Out. Then, start the next player’s turn!

(emphasis mine)
It doesn't matter whose Pokémon is asleep, it doesn't matter whose turn is next.
In your example, yes, you get to flip before your round starts (that means 50% probability that you would be stuck with sleeping Pokémon) and after your turn ends, you get to flip again.