As has been said, the landlord/agent will come after YOU @JeannieJo if the person you are guarantor for doesn't pay their rent. And YOU will end up with a default on your credit record, or bailiffs turning up at your door. (If you don't pay.) I've seen it happen. It does happen.
I think some people actually think being a guarantor is the same as giving a reference. It's not. YOU are responsible for the person's rent or debts. It's literally your debt.
I would never, and have never been a guarantor, not even for my DC. I (and DH) gave them both the first 6 months rent (for their student house at Uni,) when they had no credit rating. Fortunately, both landlords were happy with that. (Yes, I know some people aren't in the position to do this, but we were. We weren't minted but we had some savings.)
No way were we signing an open-ended agreement, to potentially end up paying multiple 1000s of pounds for any non-payment of rent or damages to the property by our DC, or anyone else living there. How fucking DARE they (many landlords and property agencies) expect people to sign an agreement to pay some other adult's debt?! That's what the jointly and severally liable agreement is that you have to sign for your DC in a Uni student house. Yeah, pretty much all parents are asked to pay it, but they will come after you all if one doesn't pay!
I found that a much better idea than signing an open-ended agreement (that would also included paying rent for my DC, or any other tenants who didn't pay, and any damages by my DC - or any other tenant.) Could have been a 5-figure sum in the end. And as has been said, the landlord will come after the guarantor if they can't get the tenant to pay.
And as @angela1952 said, no they won't come down really hard on the tenant first, they will go to them first obviously, and try to get them to pay it for a month or 2, then they will come after the guarantor, and the guarantor will be targeted for the debts. (And be taken to court/have bailiffs turning up, if they don't pay!)
It's appalling that parents are expected to be guarantor for their young adult DC. In fact, it's a fucking cheek to ask anyone to do it really! You're basically taking on someone's debt.