I don’t have a straightforward or conclusive answer, and also kanji / writing is not quite my forte, but here’s my best guess.
Short answer. It seems 吾 = 1st person singular (I) only / 我 = possible plural and reflexive (oneself). Whether this was always enforced in writing, I do not know.
Long answer. If you look up 我 and 吾 in the dictionary, you’ll actually find that 吾 is listed as the first kanji for あ（れ） and 我 for わ（れ）. But both appear as alternative kanji for both words and I have no way of checking that this usage was consistently enforced, though this would be in line with Helix’s link above (as well as a couple other Google links I could find), as I’ll explain below.
So let’s assume 吾＝あ and 我＝わ. What does it tell us? Well, first, あ（れ） doesn’t exist anymore in modern Japanese. If you hear あれ, you’ll think it’s the demonstrative “that there”, but actually it used to be a first person pronoun “I” long ago (and still in the literary language). And according to both the traditional view and modern scholarship (see for example Frellesvig’s History 3.8.2, quoted below) あ was the exclusive first person, while わ was used both for plurals and sometimes reflexively (think じぶん), though there were overlaps in usage.
Among the 1st person forms, the a- forms are only used for exclusive, singular reference (‘I, me (alone)’), whereas the wa- forms also are used for inclusive and plural reference (‘me/us (including you)’) and also reflexively (‘myself, oneself’), suggesting that the wa- forms reflect an earlier indefinite personal pronoun ‘one’. The wa- forms were used more than the a- forms and there is also some overlap in usage, indicating that a shift from a- to wa- as the 1st person pronoun was in the course of being completed; the a- forms went out of use and are not used in EMJ.
In any case, this has little influence on modern Japanese; the distinction would be considered classical at best, but you’ll find 吾 read as あ or あが in kun compounds more often, I believe. As for okurigana usage… well, classical orthography was all over the place, so particles may or may not be written explicitly. So 吾 can be あ、あれ、あが、わ、われ、わが、…
P.S.: To clarify, I am not suggesting that 吾 should always be read as あ even in classical texts; since あ as a pronoun went out of favour before Early Middle Japanese, when the literary/classical language was kind of fixed, it is very possible (again, pure speculation on my part) that some distinction was retained in the writing (kanji) long after it fused in actual word usage as わ everywhere; this is evidenced by the fact that people today still claim there is a difference depending on the kanji you use though it’s only one word now. Anyway, take my opinion with a grain of salt; as I said, kanji is definitely not my strong suit.