Search their name the way their customers type it, and read their Tagged tab, which is public. Their brand as a hashtag, the name with the space removed, their best-known product on its own, and the name as plain caption text. You do not need to follow them and nothing you do here is visible to them.
Read the accounts, not the count
The number of posts about a competitor is the least useful thing on the screen. What matters is who is making them. Open five or six of the accounts and ask what kind of person this is, what else they post, and whether they have ever mentioned you.
A creator who has featured a direct competitor twice, unpaid, is the warmest outreach list in your category, and it is sitting in public. So is the reverse signal: if everyone posting about them is the same kind of account with the same audience, the category has converged on one customer and there is room beside it.
Match the method or the comparison lies
If you are going to compare your volume against theirs, count the same way for both. A brand whose customers invented a nickname is findable by two or three strings. Liquid Death is searched as one word and two, and as the name of a specific can. A brand with a plain dictionary name, like Native, is buried under the ordinary use of the word and looks quieter than it is.
Compare a two-string brand against a one-string brand and you have measured naming, not attention. Fix the keyword set before you trust the gap.
What to write down
Once a week, five minutes: the handles that posted, roughly what kind of account each is, and anything that was not there last week. The number will bounce around and tell you nothing. A new type of account showing up three weeks running is the actual signal, and you can only see it if you kept a record.
Where the manual version runs out
Hashtag results are ranked and truncated, so you are only ever seeing recent and top. That is fine for spotting a change this week and useless for the question you will eventually ask, which is what a competitor's mentions looked like across the last three months. Searching only works forwards.
We keep that history for the brands in our catalogue, so a competitor page shows the posts and how the volume moved rather than a snapshot of today. If yours is in there, the page is public and free to read: start at the brand index and look them up. If it is not, the weekly five minutes is a genuinely reasonable substitute for a while.