@Bridget152 I had the exact same thought as @Sarah977 , which is that maybe these people who were hanging around weren't actually staying there.
We recently hosted a young family who invited some friends they know from the area over for a meal on our terrace. We have a very large terrace with a huge table that can sit 10-12 people, depending on their age and size, and we have a modern gas barbecue and a traditional Portuguese-style one. Plus, there are several deck and lounge chairs on the terrace as well.
The point is that it's a great place for doing a bit of hosting. These guests were nice enough to tell us that they were going to invite some friends over, and our neighbours told us later that they hadn't been noisy at all--just some people having food, drinks and a nice chat, and they wrapped it all up at a decent hour.
So, if your place holds 12 people, then my guess is that it is also a great place for hosting. It is quite possible, then, that your guests simply invited some friends over and that they were "hanging around" as friends.
Realise, then, that if you accuse these people of "not [being] honest about having extra people to avoid having to pay the higher rate" and you are wrong, then you are going to look bad in the eyes of this guest. So, ask yourself: is the little bit of difference in the payment that you think you didn't receive worth the risk of a bad review due to a (possibly) false accusation and messing up all of your good reviews? The way that I understand that it works with Airbnb, that one bad review will not only jeopardise your chances of becoming a Superhost, but it could also hurt you in the order that your property is listed.
Anyway, if it's still bothering you, then you might consider mentioning it politely--something like, "Oh, and I noticed that when I went to our property that there were 10 people there when you had only booked for 8 guests. Was this an oversight?" Something like that. But of course, the guest will deny it, and you won't get the satisfaction you want from asking such a question. And then, of course, you have opened up the possibility of getting a bad review.
I personally would let it go--and if you really think that this is going to continue to be a problem, then yes, install cameras near your entrance. But don't forget to state on your listing that you have cameras. Airbnb requires this, and I heard recently about a case of a host having installed cameras, not saying so and getting banned from the site, as it is apparently a legal requirement in most countries to let guests know that they are being filmed.
Good luck.