The spell description for summon monster includes this line:
"It appears where you designate and acts immediately, on your turn."
This means that the summons would act on the summoners initiative count... that turn. What about in subsequent turns?
A Summoner's (the class) monsters last minutes / level, instead of rounds, and thus are more likely to stick around for multiple battles. Would they continue to act on the Summoner's turn, or roll for their own initiative?
Additionally, the "Mighty Summons" mythic feat can grant the "Agile template" to a creature, which gives it "Dual Initiative", which seems to demand that a summoned creature gets its own initiative, independent of the Summoner's, even during the battle in which it was summoned.