Your circuit is going to damage something in short order. If you consider the transistors as back-to-back diodes (which is what bipolar junction transistors are, in a naive sense), what you have is:
simulate this circuit – Schematic created using CircuitLab
Follow the current path to see that you've created a near-short-circuit from +12V to ground via the base-emitter junctions of each transistor. I believe what you meant to draw was the third circuit "Z" here:
simulate this circuit
I've simply swapped Q1 and Q2. In my circuit it's the emitters which are tied together, and the collectors are connected to the power supply rails. In your circuit, you've done the exact opposite. The difference is subtle, but hugely important. It completely alters the behaviour of the transistors.
X and Y are emitter followers. The transistor in X is able to "pull up" the output voltage very strongly. In other words, it is able to source lots of current. However, you must rely on R1 to "pull down", or sink current, which means this circuit is good at sourcing current, but can't sink it very well without using a ridiculously low value resistor R1.
In circuit Y, the situation is reversed. This configuration is a great current sink (via Q2), and can pull the output voltage down very hard when needed, but R2 must be very small to pull up (source current) with any heft.
This is the important thing to grok; both X and Y are emitter followers, where the emitter is free to swing up and down in potential. As the bases rise and fall in potential, the transistor will "switch on" exactly enough to bring the emitter potential with it.
By combining X and Y into a single circuit Z, we are able to push and pull (sink and source current) strongly, and have an output potential that "follows" the input, albeit with an offset of 0.7V.
In your circuit you've tied the emitters to the supply rails, and since the bases are tied to their emitters with a PN junction (diode), you've also clamped the bases to within 0.7V of the supply rails. Your circuit operates in a completely different manner from mine. It's more like something you might use in a digital system, not analogue.
You really need to understand common-emitter and common-collector tolopolgies before you can embark on a project like this.