A useful explanation from Lee Airton:
If someone uses singular they you do conjugate in the plural when referring to them directly: “they HAVE an appointment.” But when using their name to refer to them, you use the singular: “Lee HAS an appointment, so remind THEM that THEY have to call ahead to confirm.”
This kind of switcheroo requires extra attentiveness when writing….[W]hen using singular they, one needs to be careful when referring to more than one person in the same paragraph (or page, etc.). Readers could interpret that you’re referring to everyone and not to the singular they user! So, using this pronoun requires more than substitution. It requires changing how we write, or at least being a bit more nit-picky!
I would add that this is true when speaking or, for example, changing the pronouns in children’s books.
For more examples of pronouns people use and how to conjugate them, check out this comprehensive Wikipedia article.
* There is currently debate about themself vs. themselves as a singular gender neutral reflexive pronoun. Consensus appears to be moving towards 1) it feels right for people who use they/them/theirs and 2) it’s is historically grammatically correct even if it sounds odd.
Third Person Pronoun Examples (modified from Wikipedia )