Vandersteen Model 2Ce Signature II

How good are these speakers? Would they work well in a 12' x 14' room using a 200W/channel solid state amp?
They are among the very best value in all of audio. They make natural music and are nearly full-range. I have 3A Sigs in a room close to yours and they work fine so I imagine the 2CE's will work just fine. I also power my 3A's with 200 watt hybrid monoblocks - plenty of headroom. You can't go wrong - go for 'em.
Great speakers. I agree with Krisjan, they are one of the best values in all of audio. I will go as far as to say they are some of the most important speakers ever built. I put them up there with Klipschorns, AR1's, Magnepan Tympani's, Quad 57's and Martin Logan CLS.

They are really the speaker that put Vandersteen on the map, and the basis for the design of all the subsequent "better" speakers from Vandersteen.
And I third the motion...
I have had mine for 2 years now, running them with an Anthem 200 wpc solid state amp and Rogue Magnum tube pre. Great sound in a fairly large room
Absolutely, they would work well. Mine are in a room slightly bigger with a Classe amp.
Are these speakers "picky" when it comes to room placement? That is, how far from the back wall and side wall should they be?
Oops, I forgot to mention, can they be bi-amped?
Yes they can be bi-amped or bi-wired. Vandersteen web site has some detailed discussion of those issues.
Give it or take there are no equal ears around there. Heard them (aiming to upgrade my 1Cs bunch of years ago) with 50Wpc tube amp and heard them with 200W/ch SS amp. In both cases they're too slow for rock and instrumental music I listen to. They also required to be standing on matching Sound Anchor stands which would make you to spend another $$couple. If you buy them used than most-probably the stands will be included.
I know you are asking about the "2" but the "1" could not breath in my 11 x 10 room. I understand my room is smaller than yours but the "2" is bigger than the "1" and you'd be well advised to try a stand-mount first. All I'm saying is be careful.
IMHO, perhaps the greates value in high end audio, they are fussy about room placement and your room might be a bit too tight.
Damned good! Regarding power, I drive mine with Ayre AX-7e (60 watts) to great affect tho' I cannot reproduce concert levels of The Who. I love the combo. Regarding room size, mine is considerably larger (18x19' with a vaulted ceiling) but I have heard 3A Sigs in a room about your size and I thought it sounded fine, excellent, in fact. IMO, you need to keep them away from the front (and sides) wall by >3' and also need to be able to sit about 8' away. In my room, the listening position is against the rear wall tho' I also have an opening to the next room behind me. When I heard those 3As, the listening position was also against the rear wall with the speakers about 3' from the front wall. Good luck.
I think they are a solid value and good all around performer but, they have a lot of competition and it depends on what you want a speaker to do. I would prefer a Reynaud Trente in that room for instance (even though the Vandy is more of a full range speaker).

Just food for thought...
I don't have the "Mark II" version of the 2Ce Signature, but I use my 2Ce Sigs in a room that's a hair over 11' X 15'.

They have probably 4 feet distance from the side walls with very little distance to the wall behind them, and they definitely sound outstanding.

These speakers love to be driven with powerful gear, so you'd definitely benefit from using a beast of an amp (I drive mine with a Plinius).

Supposedly the Mk 2 version doesn't have the traditional Vandy house sound, so just keep that in mind. A secondhand pair of the regular version will be an amazingly good deal.
A couple of notes for disussion...

1. In a room of your size, you'll probably also want to invest in some acoustical panels. This is regardless of speaker choice. No, it's not a sexy way to spend money, but it's some of the best money you'll spend. The 2's put out a lot of low end and will (probably) overload the room without treatments.

2. Can the 2Ce Sig II's be bi-amped. Yes. Should they be? Well...that's up for debate / discussion. Is really depends on the amp and if it can handle the predominantly capacitive load of the mid/tweeter crossover. In general, Vandersteen himself advises against it (don't take my word for is, go out to the Vandersteen website and check for yourself).
Bi-Wire, not Bi-amp