is saying this right:

theirs

That house is theirs.

is this sentence correct

either alan or john will collect signatures for THEIRS petition on tuesday

or is it HIS

The correct word in this sentence is HIS.

The construction of "either . . . or" is singular so a singular pronoun is used.

However, if the subject was plural and said, then --

Alan and John will collect signatures for THEIR peitition on Tuesday.

is it his

Yes.

Yes, the word "theirs" is used correctly to indicate possession by multiple people or entities. It is a pronoun used to show that something belongs to a group of people, typically used in the phrase "It is theirs."

To determine if a word is used correctly in a sentence, you can consider the context and the grammatical rules. In this case, "theirs" is a possessive pronoun which is used to show ownership. As long as the sentence contextually makes sense, and the word "theirs" is used in the appropriate place, then it is said correctly.