In some languages pronouns are often optional. For example, in Spanish you might say "estoy feliz" for "I'm happy" though it literally translates to "am happy". In both Spanish and English the implied pronoun is clear—'estoy' and 'am' are first person singular conjugations—but in Spanish it's grammatical to leave out the pronoun while in English it's not. Languages vary in whether they allow you to leave it out, and those that do are called pro-drop. [1]

Because pronouns in English are traditionally gendered and we haven't finished switching over to singular 'they' yet, sometimes you're in a position where the sentence in your mind calls for a pronoun but you don't know what pronouns the person in question prefers. In some of these cases speaking as if English were pro-drop can be helpful.

For example, say you're talking to someone, and then they go off to the bathroom to brush their teeth. A friend of theirs comes along and asks where they went. What comes to mind is "PRO's off brushing PRO's teeth," but you don't know whether it's "he's off brushing his teeth," "she's off brushing her teeth," or another pronoun. Normally I'd say "they're off brushing their teeth," but another option here is to leave out the pronouns entirely and say "off brushing teeth". This sounds weird on its own, but in context it's fine:

A: Where's Leslie?
B: Off brushing teeth.

In speech this kind of casual pro-drop is already used in some cases, so instead of introducing something completely new we're just talking about pushing a little into situations where it's not the form people would typically use. This avoids drawing the conversation into "what did you just say?" while leaving gender out.

(If you need to say more than a sentence or two, though, the person you're talking to will probably notice you're avoiding pronouns and you might get sidetracked.)

A few more examples:

A: How do you know Eli?
B: Went to my school; was in a lot of my classes.

A: Was there anyone new at the meeting?
B: Pat was there. Engineer visiting from Philly; a bit of a jerk really.

A: I just can't stand Jean.
B: Oh? How so?
A: Just bugs me. Always interrupting.
B: I don't know, honestly I think you interrupt people more.

The key thing is that this only works in cases where it really is obvious to your listener which person you're talking about.

(In general, still make an effort to learn what pronouns people use, and use their preferred ones consistently. If you find yourself slipping into a pattern where you're using singular they, singular you, the substitutions, and/or pro-drop for someone you refer to at all regularly then you should take some time to practice their chosen pronoun on your own. But there are still going to be situations with new people where it's just not the right moment to get into pronouns, and there these strategies can be positive.)

[1] Spanish isn't really the ideal example here since it generally only allows dropping subject pronouns, and so is sometimes considered null subject instead of pro-drop. But I know Spanish better than any of the more fitting languages.