I stayed in the hostel, and have no experience with the flatshares.
Have stayed in the Dover Castle 3 times, and for 3 nights on each occasion. I've been in the same room each time.
For what it is, this is a perfectly fine hostel. It's well maintained, there's a common room/kitchen area with television, oven, sink, seating, and a table, as well as a fridge. There's also plates etc provided, which are your responsibility to clean. I happened to be in there at 6am one Monday morning, when the cleaner came in. Can't say a lot about his politeness skills, but at least it shows the place is cleaned. Showers are fine. They tend to be warm, but get hot if you stay in too long, and then take a long while to cool down (there is no temperature control). It can be quite hard to get in to the bathroom, since on the floor I stayed there was one bathroom (Oddly, there's two showers inside, partitioned by separate shower curtains, but one door and lock - so maybe arrange to shower the same time as a friend). There is also only one toilet, but there are toilets downstairs in the pub so that's not really an issue.
The volume outside can't be controlled. London's a busy city, there'll be noise regardless as to how fancy your accommodation is. Thursday through to Saturday the bar downstairs is open late, so if you're on the first floor you'll probably get some music and chatter sounds. I've never been on the upper floors, so cannot vouch for how different the noise is up there. I've never had a problem sleeping there, though, and I am a troubled sleeper. The bar downstairs IS very good, though, and the staff have been friendly everytime I've been there. There's a pool table, and a good music selection to chose from.
Security is good, there's a lock to get in, and a lock to enter the bedroom area, then each dorm has a key. I seem to recall there being additional storage for valuables, downstairs, but I've not used it so I'm not sure. Luggage storage is next to reception.
I've never personally had anything stolen there, but I suppose it's luck of the draw as to who's in your room, so I'd suggest making sure you have a good lock on your suitcase. Basically, exercise general caution.
The beds were comfy enough, and the linen was clean. There are plug sockets in the room, under the beds. The beds are bunk-beds. I don't think they have ladders, but it's easy enough to get up !
The hostel is incredibly convenient for the underground, it's no more than a minute from Borough station. I usually arrive on the train at Waterloo and walk, since it's only a fifteen minute walk from there.
I'd have no problems with staying there again, but I think it's better the more people you travel with. If you can secure a whole dorm to yourselves, then I'd certainly recommend that. Would not recommend for business travellers, or families with young children.