AWG gauge for 10W cob leds and common ground
IEC recommends 30 degr. Cercius as a maximum temperature rise on a wire, under normal conditions.

You have a return of a total 2.7A. So the thinner wire that you can use is AWG24 (0.0511cm diameter) insulated straigt wire standing at free air.

Insulated AWG22 (0.003257 cm^2 cross section) will rise a temperature of 15 degr. Celcius @3A load under same conditions.

If you have to use a long wire, check the voltage drop as well.

EDIT
to answer additional questions

IEE set maximum acceptable drop on conductors the 2,5% of the nominal input voltage. 
In case of DC apply Ohm's law to calculate Voltage drop i.e. :

                                 **V=IxR**


I is the total current of your load(s)

R is the total conductor resistance.

                                **R=ρxL/A**


ρ= resitivity of the conductor. For copper at 25degr. Celcius is 0.023 Ωmm^2/m (or from manufacturers tables Ω/km for the specific AWG#)

L= the TOTAL length of the conductor in meters

A= the conductor cross section in mm^2

For AC some additional factors sould taking into account.

Having resistance and current, you can start calculating the power dissipation from conductor and using thermodymnamics to calculate temperature. There is a big difference between bare conductor (or PCB trace) and insulated one. 

Common conductor (return) calculated using the sum  of all currents.