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The word θάλασσα thálassa "sea" is spelled in various different ways, with different letters replacing the sigmas: some dialects had a tau, for example, while others had a theta.

Do we know (through loans and cognates, for example, or transcriptions into other languages) what underlying sound these various letters were representing?

There have been various theories about the phonetic value of ττ and σσ, but it`s often held that they were pronounced as might be expected, i.e. as [tt] and [ss]. The philologist Sidney Allen argues as follows:

These facts have led some scholars to suppose that both the ττ of
Attic and the σσ of other dialects represent different attempts to
write such an affricate without the use of a special symbol; and that
the pronunciation as a double plosive or fricative is a post-classical
development, based in part at least on the spelling. But apart from
the improbability of spelling influence on colloquial speech in
antiquity, it is scarcely credible that the existence of an affricate
sound would not have been revealed in any inscriptional spelling
outside those mentioned above (e.g. as τσ), nor the tradition of it
survive in the account of any grammarian. On the other hand it is
perfectly feasible for both [tt] and [ss] to develop from an earlier
affricate, and there seems therefore no need whatever to assume that
the ττ of Attic or the σσ of other dialects mean anything more than
they appear to. (Sidney Allen, Vox Graeca, A Guide to the Pronunciation of Classical Greek, pg.58)

Geoffrey Horrocks also attributes the values [tt] and [ss] to ττ and σσ.

[I]t was noted that many of the adopted place names and vocabulary
items borrowed from the pre-Greek languages of the Aegean basin had
undergone dialectally diagnostic sound changes. The almost certainly
borrowed word for ‘sea’ , for example, has the following forms:
both of which reveal the
dialectally standard products of the palatalization of an original
voiceless dental or velar by a following semi - vowel. 3 Consider the
example in (6): Allen (1958)
explains this divergent dialectal development on the assumption of a
generalized heavy palatalization of /t/ in Boeotian: the Attic reflex
is then probably due to close contact with Boeotian at the time of the
change (on which see further below). But the fact that loanwords such
as that in (5) undergo developments identical to those undergone by
native vocabulary (even though we cannot, of course, discover the
exact form in which such words were first borrowed) strongly suggests
that the division of Greek into the historical dialects attested in
literature and alphabetic inscriptions had only taken place after all
its future speakers had become established in the Aegean area.
(Geoffrey Horrocks, Greek, A History of the Language and its
Speakers, pg.19)

Additional information:

Although your question doesn't deal with the following, I thought you might be interested:

But, like many literary languages, literary Attic was subject to
influences from outside the restricted area of the spoken dialect,
most particularly from Ionic. And one of the most characteristic
features of this influence is the substitution of forms with σσ for
the ττ of 'pure' Attic as exemplified by the inscriptions. In fact in
tragedy, and in prose works up to and including Thucydides, the ττ of
Attic is almost entirely avoided. Even though normal Attic grammar was
used, and Attic phonology generally adopted, it seems that the ττ was
felt as something of a provincialism by contrast with the σσ of most
of the rest of the Greek-speaking world—all the more to be avoided as
a characteristic of the speech of the 'συοβοιωτοί'; and even false
Ionicisms (notably ἡσσᾶσθαι as against Attic ἡττᾶσθαι and Ionic
ἑσσοῦσθαι) were liable to be perpetrated in avoidance of this
shibboleth. (Sidney Allen, Vox Graeca, A Guide to the Pronunciation of Classical Greek, pg.11)

We can only speculate about the exact nature of the "foreign phoneme" but everyone agrees that θάλασσα is Pre-Greek (i.e. not IE), one of the reasons being that there were no geminates in neither in the PIE nor in Pre-Greek (Beekes, Brixhe etc.).

As Stephen Colvin (Colvin 2007) writes, "the prehistory of these clusters [i.e. obstruent +y, Alex B.] is complex and much disputed" (p. 26).

Bubenik 2017 offers a very clear and a rather compelling account of how this might have happened. He writes that dental and velar palatalization, with subsequent affrication, happened in Proto-Greek:

*tj > *t'j > *t's'j

*kj > *k'j > *t's'j (in his notation).

The palatal glide was later lost, the palatal affricate was depalatalized and merged with Proto-Greek *ts.

This cluster, Bubenik writes, "could be subject to progressive assimilation ts > tt (in Boeotian, Attic and Central Cretan) or to regressive assimilation ts> ss (in other dialects)' (p. 647). Thus, he classifies all the dialects into the following groups:

1. Arcado-Cretan and Ionic: PG *k(h)j, tw > ss; PG *t(h)j, *ts and *ss > s;


2. Aeolic and West: all of those > ss;


3. Attica, Euboea, and Boeotia: PG *k(h)j, *tw, and partly **t(h)j > tt.

He speculates that tt "could have belonged to the Aeolic basilect, surviving in Boeotian (and extended to [Western?] Attic), but eliminated partially in Thessalian and wholly in Lesbian" (p. 648).

Also, it seems that almost everyone believes the underlying consonant to be a palatalized velar, e.g. *-χyᾰ (Lejeune 1972, §98d) or *kʲᾰ (Beekes, kya in his notation); cf. Macedonian (?) θαλάγχα(ν).

cf. "geminate tt in Attic is a reflex of part of the palatalization isogloss shared with Boeotian and Euboean, corresponding to the geminate ss of Ionic and other dialects: cf. lexical forms like thálatta ‘sea’, glôtta ‘tongue’ vs. Ion. thálassa, glôssa, or verbal formations like *eret-jō > eréttō ‘I row’ (cf. erétēs ‘rower’), *kāruk-jō > kērúttō ‘I announce’ vs. Ion. eréssō, kērússō" (Angeliki Malikouti-Drachman, “Phonology (Survey)”, in: Encyclopedia of Ancient Greek Language and Linguistics, 2013).