I recently found 言いあったり in one of my sentences, and I can't seem to find what it means.
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What does 言いあったり mean?
This is a conjugated form of the verb 言【い】い合【あ】う.
So 言【い】い + 合【あ】う + "to say to each other".
As for what 言【い】い合【あ】ったり means with the conjugation included, we look at the second part of your question:
... and how is it conjugated?
言いあったり has the ～たり conjugation, which is used to refer to a non-exhaustive list of activities. This is often taught and encountered in the format:
It is also perfectly fine to use a single verb with the ～たり conjugation, as we see in your sample sentence. This gives the sense that the subject does that one verb, plus other things that are only implied but not mentioned.
To conjugate a verb into the non-exhaustive ～たりform, the basic rule is to take the past tense of the dictionary form of the verb and stick り on the end. In compound verbs, we only fully conjugate the last verb in the compound. In 言【い】い合【あ】う, the last verb is 合【あ】う. The past tense of 合【あ】う is 合【あ】った, so we can make the ～たり conjugation by adding the り, producing 言【い】い合【あ】ったり ("saying to each other, and doing other things that aren't mentioned").
Please comment if the above does not fully address your question, and I can edit the post to update.