OK, I just wrote out a massive reply, and accidentally hit 'home', and lost it all....
Firstly, mechanical switches switch pretty damn fast - from almost infinite resistance, to very low resistance very quickly. In this circuit, it seems at 'idle' the supply charges up a capacitor... once full the cap has massive resistance, so really doesn't drain the power at all. When the switch is pressed, the capacitor suddenly discharges through the coil, (which is probably a camera flash trigger transformer?). Any resistance in the path of the switch slows down the discharge, giving much less voltage on the output = little, if any, spark. A single transistor if somehow turned on instantly, still might not be able to switch the high current spike fast enough.
So you need something that can take a rather large amount of current, for a very short period of time as the cap dumps all its energy across the coil - mechanical switches/relays are ideal. But solid state should work
Either a power transistor (PNP), or a Pchannel MOSFET.
Some idea's:
As suggested, darlingtons are probably too slow, and would reduce the voltage from the cap to the coil (higher saturation voltage than a single transistor). So, maybe drive the main transistor with a JFET? - those are often used in touch sensitive circuits. Or perhaps a logic IC. CMOS logic (CD4000 series) would be ideal as they can run at 12V, where-as a 74HC series needs 5v, and so would require a regulator (as would a PIC micro-controller ...which is over kill!)
I'm sure a single transistor is capable or switching the cap (1 to 2 on your schem), but that transistor needs to be saturated quickly.
As it switches from a charged cap (+12V) to the load (the coil) a PNP should be used. To turn that on, its base needs to be pulled from 12V, to <11V via a resistor. An NPN (or N channel MOSFET) could be used for this, although it might not have enough gain to turn the PNP on fast/hard enough for it to switch that current. - maybe an NPN darlington for this. Thats a total of three transistors (darlingtons can be in one package, so thats only two devices). Again, and FET could be used, but like the CMOS logic would need input protection.
This page has loads:
http://talkingelectronics.com/projects/TouchSwitch/TouchSwitch-1.html
I'll admit, I've never seen a lighter sparker that complicated, its mostly piezoelectric, or flint.... or at the other end of the spectrum, for oil burners, 25-50kV
Is this a hand held thing? cigarette lighter? with a small 12V battery? if thats the case, then I'm sure a power MOSFET is overkill - not to mention too big, as would a power transistor. So many options! If you've got a pic of what you're dealing with, space requirements, it should narrow down whats easiest/best. I still say, best case, couple of transistors - worst case, a transistor and a small logic chip