The best value for money accommodation in Sochi is private rooms, sometimes in the owners home, sharing the facilities with them, or else one where the accommodation is solely yours and the owner lives nearby. Commonly elderly ladies rent out their flats and then live nearby with friends or family. This brings them much needed income.
It's important to know that if you are just arriving in Russia, most likely having sailed from Georgia you do need to register your visa; the office is near the train station but is not open daily and operates quite erratic hours on those days it is open. The most hassle free way of dealing with this is to spend your first night in a hotel and have the staff register your visa. Once this is done, get yourself down to the train station and get some private rooms.
You will not need to look for prospective landladies. They will find you; they can sense you from some distance away. The trouble is most speak no English and since you are unlikely to speak Russian - I speak a little but I wish I'd known more - you need a translator. Fear not - a lady who speaks basic English translates what each of the landladies is saying. If you tell her what you want she'll translates that back into Russian and ask which landlady can supply that.
For example, the translating lady will say 'This lady has a flat with two rooms, kitchen and balcony' or 'This lady is offering one double room in her house, it's near the beach'. Having found something that sounds good you then need to negotiate a price - this is only your first negotiation and should be used to test the water since you still have to see the flat and this could involve a taxi ride (if you are paying she won't object) or a bus ride. We followed our landlady and scrambled onto a bus that hd almost departed the bus station!
The bus was packed but she still managed to nudge us and point out various landmarks, presumably to demonstrate the handy loction of her flat. It was a large bedsit with a separate kitchen and bathroom in a small block overlooking a leafy square. It was beside the bus stop to get into town and close to shops, restaurants and an internet cafe. The kitchen was well equipped and there was loads of clean linen.
On the whole private accommodation in Sochi is of this standard but of course you shouldn't hand over any money unti lyou've checked out the accommodation. Similarly, don't make the mistake of saying yes because you've gone a long way and can't be bothered to look elsewhere. Poor accommdation can put a serious downer on your trip so don't feel obliged to take it if it's not worth the money.
Assuming all is well get your landlady to demonstrate anything that might be tricky before she leaves you to it (if she doesn't live there) - that includes water, heating, locks, windows, etc. And do get a telephone number - you may not be able to talk to her in Russian but you may be able to show a neighbour what the problem is and ask them to phone her on your behalf.
Aim to pay around $10 per night for something self-contained without the owner living there but never be mean and do consider what the accommodation is worth to you and in comparison with a rather more costly night in a hotel.