This is a question both landlords and tenants seem to ask quite a lot so here are my comments.
It depends on the type of tenancy you have.
If the tenant is renting a room in a shared house
By this I mean if the tenant has his (or her) own tenancy agreement for their own room and share the rest of the property with the other tenants – all of whom also have their own tenancy for their own room, etc
Then yes, the doors should have locks on them. The room is the ‘property’ that the tenant is renting and as this is a tenancy, they have the right to keep everyone out. Including you.
So the room has to have a lock.
If the tenant is renting a house or flat jointly with other tenants
By this I mean if all the tenants have signed one agreement, together. Here it is really up to the parties what they want and what they agree.
So far as the legal situation is concerned, the tenants are renting the whole of the property together. So they all have an equal right to use all the rooms.
Of course they will almost certainly each have their own room in the property and may want to keep their fellow joint tenants out of it.
But they don’t have a legal right to require the landlord to put locks on the doors, if they are not there already. It is up to the tenants to sort this out themselves.
If asked, the landlord may or may not give permission for the tenant to put their own locks on the doors.
Although if the landlord refuses permission, there is not really anything they can do to stop the tenant putting a lock on the door anyway, at their own expense.
However if this damages the door in any way or if the landlord really does not want the lock there, the landlord will normally be entitled (if the lock was added against their wishes) to claim for the cost of getting the lock removed and the door made good from the deposit at the end of the tenancy.