Bridging transistors
No, connecting the emitters together as you show will not cause a short circuit. However, it is unlikely that the current supplied to the LEDs will be shared equally by the two transistors. Small differences between the transistors will cause one of the them to supply most of the current. Usually, when transistors are placed in parallel like this, a low-value resistor will be added at each emitter to provide some negative feedback and equalize the currents.
Also, your two LEDs will probably not share the current equally if you connect them in parallel. If they are significantly different, say one red and one blue, then you may see that only one of them lights up at all.