In that sentence, you don't need any apostrophes.
As you probably know, the construction apostrophe+s is a way to build the genitive case of nouns in English. This is a remnant of an old case system that has been largely lost in modern English.
According to CGEL, there are six types of genitive construction (p. 467):
 i [Kim's father] has arrived. [I: subject-determiner]
ii No one objected to [Kim's joining the party]. [II: subject of gerund-participial]
iii Max's attempt wasn't as good as [Kim's]. [III: fused subject-determiner-head]
iv She's [ a friend of Kim's]. [IV: oblique genitive]
v All this is Kim's. [V: predicative genitive]
vi He lives in [an old people's home]. [VI: attributive genitive]
But in your sentence, the noun headphones appears as the head of the noun phrase (NP) separate headphones (separate being an adjective, a pre-head modifier of headphones); and the NP separate headphones, in turn, appears as the direct object of the verb own. We see that the function of the direct object is not one of the six genitive constructions listed above.
Instead, the direct object function is normally associated with the accusative case. In English, the accusative case survives only for some of the personal pronouns and the relative pronoun who: (I → me, he → him, she → her, we → us, they → them, who → whom). This is why we say I saw her (but not *I saw she) and She saw me (but not *She saw I). In particular, you would say She owns me (me = the accusative of I), not *She owns I (I = the nominative of I) or *she owns my (my = the genitive of I).
But nouns like headphones do not inflect for the accusative case, and so, when functioning as direct objects (or as heads of NPs functioning as direct objects), they are kept in the plain case.1
1The accusative case is normally contrasted with the nominative case, which is the case the noun has when it appears as the subject of a finite clause (i.e. not as the subject of a gerund-participal clause). But since for English nouns there is no distinction between the nominative and the accusative, CGEL wants to 'neutralize the nominative-accusative opposition', and therefore refers to the nominative/accusative form of the noun as the 'plain case' of the noun. In contrast, for personal pronouns, CGEL retains the notion of the nominative and the accusative.
Incidentally, in many cases, you can use the behavior of pronouns to figure out if you need to put your noun in the genitive. Just replace the relevant NP by a personal pronoun. If it turns out that the pronoun must be in the genitive (my/mine, your/yours, his, her/hers, its, our/ours, their/theirs, whose), then you need to place your head noun in the genitive. Otherwise, your head noun should remain in the plain case.
Having said that, you do have one genitive in that sentence: their. This is the genitive of the noun they, and it is not formed by adding 's but by this more substantial change.
Here are some constructions where you would use the genitive of the nouns:
Sophie's, Brock's, and Hannah's headphones
the three iPhones' headphones (= the headphones, probably multiple, that belong
to the three iPhones; grammatical, but probably
few people would actually put it like that)
Sophie's iPhones's headphones (again, grammatical, but probably few people would actually put it like that)