How does Persistent Spell work with spells that have a duration?

Question

How does persistent spell work with spells that have a duration? The ones that I am most interested in are constricting coils and black tentacles.

Persistent Spell

Whenever a creature targeted by a persistent spell or within its area succeeds on its saving throw against the spell, it must make another saving throw against the effect. If a creature fails this second saving throw, it suffers the full effects of the spell, as if it had failed its first saving throw.

Constricting coils

Duration 1 round/level

Black tentacles

Duration 1 round/level

Now the coils spell would normally end once you made your save, but from my understanding of the metamagic, you would now be forced to save against the spell again. Would this keep happening until the spell ends each time you made the first save but not the second, or would the metamagic feat only allow the spell to continue once?

As for black T, it remains until the spell ends. How exactly does the metamagic feat work on this one? Does the grapple check count as the saving throw in this case?

Answer 1

Persistent spell simply allows a spell to invoke two saving throws instead of one. If the target fails either of the saving throws, they suffer the full effects of the spell unless they have the ability Evasion or Improved Evasion (in the instance of spells requiring Reflex saves)

In the instance of Constricting coils, a Will save is triggered and if failed the target is subject to Paralyzation and constriction. Since the spell is persistent, whenever a will save is attempted (at the beginning of that round) two saves must be rolled instead to determine if the target can break free of the paralysis/constriction.

In the instance of Black Tentacles, that spell does not have a saving throw, therefore it is not subject to being Persistent. At the bottom of the Persistent spell metamagic feat, it reads:

Spells that do not require a saving throw to resist or lessen the spell’s effect do not benefit from this feat.

Answer 2

For a persistent spell, if it requires multiple saving throws, each one will require two rolls.

Other rolls don't "count as" saving throws. Saving throws are saving throws. A grapple check isn't a saving throw, an ability check, an Escape Artist check, a skill check... None of those are saving throws.