Just to illuminate the flip-side:
We rent our flat in a private agreement with the landlady. We advertised ourselves as tenants on Gumtree because we were sick and tired of overpricing and nonsense from agents. As well as some obvious scammers, we had a few genuine replies and offers of really decent properties from landlords who likewise had got sick of dealing with agents. They stick 8% on for the landlord and take another 10% off the tenant it seems, so it ends up being a very expensive way to rent and the service you get is so often still utterly shit.
Our landlady asked for employment, landlord and personal references and got a credit check, which she paid for. As we are both self-employed we also happily provided a guarantor.
We have found it to be a wonderful arrangement and our landlady describes us as 'perfect tenants', and we've just renewed the contract for a further year.
My advice would be that you're more likely to be able to get a good feel for your tenants if you find them directly, and there are loads of good, nice, clean, honest tenants out there who want to look after a home, and are equally sick of being done over by agencies (£150 for a credit check that costs £18 online, anyone?). But trust your instincts and if there's anything that seems amiss with them, don't risk it.