3 wired old light switch to a new light switch
In your old switch I expect the middle wire is the permanent live (hot) and the left and right wires each go to one of the two lamps. You can see that the left wire is not actually connected to the upper switch.

This is a system of wiring lights known as a "switch loop". In this system there are no neutral wires running to the switch. 

You don't show any ground (earth) wires either, which would be very strange where I live but may be have been normal in the past where you live.

You can check which wire goes to which lamp by turning off the circuit breaker, checking the circuit is dead and then undoing the right wire. Make it safe by applying electricians tape or by inserting it into a loose wire-nut or terminal block. Then you can turn on the circuit breaker and see which one of the two lights no longer turns on.

The new switch should have come with connection instructions. If not, I would use a continuity tester to find the common connections and the switched connections in the new switches before installing them.

On closer inspection, I see there is a connection diagram moulded into the back of each switch



I would wire it like this:



Note that you need to cut the middle wire shorter and you need a new insulated piece of wire to connect the "common" terminals of the two switches

The new switches also have an insulation-trimming guide moulded on the back, which you should use if possible.

If the new switch is a smart-switch of some sort it probably needs a neutral connection. In that case you need to call an electrician to run new wires from a junction box to the switch. The electrician may need to make holes in your walls ceiling or floors which will need repairing and redecorating.