According to Volo's Guide to Monsters, beholders reproduce by dreaming of beholders. Do all beholder types, like gazers and spectators, also procreate this way?
When beholders dream of beholders, that's when the real trouble starts.
Beholders create other beholders and beholder-kin through dreaming, as described in Volo on pp. 6-7.
None of the beholder-kin's descriptions mention anything about them procreating. It seems reasonable, then, to conclude that all beholders, spectators, gazers, death kisses, and gauths are born of beholders and that all the various kin don't procreate, themselves.
Here Gygax states that beholders may use parthenogenic reproduction to reproduce and that no eggs have ever been found.
Gary Gygax in Monstrous Manual - 1993
Beholders may use parthenogenic reproduction to duplicate themselves, and give birth live (no beholder eggs have been found).
He also states that beholders do have "hive mothers."
Beholders and beholder-kin are usually solitary creatures, but there are reports of large communities of them surviving deep beneath the earth and in the void between the stars, under the dominion of hive mothers.
Here he mentions that some people claim they can mate with other beholders.
Beholders may also (rarely) mate with types of beholder-kin.
I think these details are purposefully vague to add an element of mystery to beholders.
Pure conjecture here
If you were to amalgamate all of these ideas together, it seems lone beholders find each other in another plane while they are sleeping and "mate". Then they produce a live birth from this process.
Gygax explicitly states that all beholder-kin are actually the same "species" and that they all reproduce in the same method, so it would seem that spectators are "born" the same way.
According to 5e's Monster Manual, Spectators are summoned monsters.
A spectator is a lesser beholder that is summoned from another plane of existence by a magical ritual, the components of which include four beholder eyestalks...
No word on how they breed, though.