The letter Z was borrowed into the Latin alphabet in order to transcribe Greek loanwords, along with Y.
Presumably, educated Latin-speakers pronounced it like its source, Greek zeta. However, Greek zeta had various different pronunciations across time periods and dialects.
What can we say about the Latin pronunciation of Z in loanwords? Do any ancient grammarians comment on this, for example? Or can transcriptions of Greek words into other languages confirm its pronunciation at that time?
Two reasons for thinking that Z was pronounced in Latin as a fricative:
The spelling SS was once used to represent it, as you mention in your prior question When did the Romans start using Z?
The spelling "ZM" existed as a variant for "SM" in words from Greek with ΣΜ. Since Σ is not thought to have been pronounced as an affricate in Greek, presumably, these words with ZM spellings weren't pronounced with an affricate in Latin either, but with a fricative. Phi Examples (e.g. Zmyrna, zmaragdos)
Both of these are consistent with a pronunciation of zeta as either [z(ː)] or [s(ː)]
On the other hand, the use of Z as an alternative for DI in some inscriptions can be seen as evidence that Z could potentially represent an affricate.
