You take ongoing damage at the start of your turn. You make saving throws at the end of your turn. "You" means the person afflicted by the effect, not by the attacker who put the effect on you. So for minions, any ongoing damage would probably kill them outright (most low-level minions die when they take 1 point of damage).
From the DDE pregen wizard (link on main page):
Sleep
Daily - Arcane, Implement, Sleep
Standard Action - Area burst 2 within 20 squares
Target: each creature in burst
Attack: +5 vs. Will
Hit: The target is slowed (save ends). If the target fails its first saving throw against this power, the target becomes unconscious (save ends).
Miss: The target is slowed (save ends).
So, the combat would go like this:
1) Wizard's Turn: casts sleep and rolls +5 vs. Will save. He hits and affects 2 minions (minion A and minion B), out of 4 total.
2) Minions' Turn: All of the minions are slowed. They all take their actions. At the end of their turn, they roll d20. If they roll 10 or higher, they made their save
Minion A makes his save: He is no longer slowed. He no longer has to make saving throws.
Minion B fails his save: He becomes unconscious. Each round, at the end of his turn, he rolls a saving throw to see if he wakes up.
Minion C makes his save: He is no longer slowed. He no longer has to make saving throws.
Minion D fails his save: He is still slowed. Each round, at the end of his turn, he rolls a saving throw to resist the slowed condition.
Finally, note that from the orcs preview that "a missed attack never damages a minion." However, I'm reading this as immunity to damage (whether ongoing or immediate). I'm assuming that minions can still be effected by other status effects from missed powers (such as the slow effect of a missed sleep).