To stick with the distinction between possessive pronoun vs. determiner, consider this excellent description offered by Cambridge Dictionary:
We use pronouns to refer to possession and ‘belonging’. There are two types: possessive pronouns and possessive determiners. We use possessive determiners before a noun. We use possessive pronouns in place of a noun
They provide this example:
Is that [determiner] your scarf? It’s very similar to [pronoun] mine
They also offer this advice with an example:
Don’t use possessive pronouns before nouns
Lots of our friends were at the party.
Not: Lots of ours friends …
It's understandable that the two can be confused, since they can both replace a Proper noun. Consider these examples:
The apple is yours. (possessive pronoun)
This is your apple. (determiner)
In either case, the word could be replaced with "Alice's."
The apple is Alice's.
This is Alice's apple.
The structure of the sentence in the question takes the form of the latter, where "your" functions as a determiner. Remember, as Cambridge recommended, you don't use a possessive pronoun before a noun. In the example given, "answers" is a noun, so you would use a determiner, not a possesive pronoun.
In response to your answers
In response to yours answers
The fact that you are adding another person to the equation doesn't affect this use (with the exception of a caveat I'll get to in a moment). You still would write
In response to your and Alice's answers
The caveat is, as was pointed out in the other answer, this is still a somewhat awkward phrasing, and it could be rephrased "In response to you and Alice's answers," just like you might write "In response to Steve and Alice's answers." But if you want to adhere to the structure in the question, you can't go wrong following the guidelines provided by Cambridge. Since "your" modifies "answers," it functions as a determiner.