The sensible answer is that a landlord should do all the checks that an agency would normally do on their behalf — but there's no need to worry because these checks are quite easy to do.
First of all, ask the tenant for proof of ID (passport, driver's licence, NUS card) so you know they are who they say they are, plus a reference from their current landlord that should include the address of the property so you can check on the land registry that the person offering the reference is indeed the owner. It's wise to get a phone number for the current landlord too, so that you can call them and check all the details in any written reference.
I know what you're thinking — a tenant could give you the number of a friend posing as their landlord. Yes, they can, but you might be able to sniff out a bogus landlord if you probe a little.
Besides, short of hiring a private detective to check on tenants, there's a limit to how thorough you can be.
You should also ask for copies of the tenant's bank statements for the last three months so you can see if they're solvent, if they've made regular rent payments, and if their salary goes into their account.
You could also use the statements to double-check the address they gave you. Finally, you should ask for a reference from their employer stating their salary and the terms of their contract, so you can see if they're likely to have a job for the duration of the tenancy.
If you really can't be bothered to do all of the above, you can pay a tenant referencing agency to do something similar for you. A detailed report on a tenant, including a credit check, costs £25 to £40 depending on which agency you use. Experian is one, but there are lots online.
You need to ask the tenant's permission to run a check and you could ask them to pay for it — most letting agents do. Certainly, if they cough up you'll know that they're serious about taking the property and they've probably got nothing to hide. If you feel a bit stingy doing this, you could say you'll refund the cost if they take the property.
Now then, that's the sensible answer, but you have to remember that none of the above will guarantee you a great tenant. Sure, some of the checks should weed out the real rotters — I doubt that anyone planning to operate a terrorist cell from your living room would be willing to hand over their passport and bank statements — but a tenant who ticks all the boxes can still turn out to be a pain in the neck.
I had one tenant who looked perfect on paper. She had a sensible job, decent salary, owned her own property.
She was also reckless and inconsiderate, and a pathological liar, but that wasn't flagged up in any of the reports. On the other hand, I've taken on tenants without references of any kind and they've been excellent.
So now I've developed a test of my own for all future tenants. I sit them down with a cup of tea, look deep into their eyes and see if I can trust them. Some might find this approach a touch spooky, but honestly, it works for me.