Bypassing Push Button with Arduino

I'm new to this, so if there is some obvious way about this I've missed, feel free to restart my approach:

I have a little push button transmitter that controls a device in the house. A picture of this circuit board is attached. Pushing either of the buttons on it accomplishes my task. I'm trying to make it activate when motion is detected.

So, on my Arduino I have a PIR sensor that is correctly detecting motion. I'm trying to get the Arduino to "push" that button when motion is detected. How would I bypass the button on the other device using the Arduino? Is a transistor of use here? I know I can short 2 of the terminals on the button and that does send the signal to the receiving device, but I'm unsure how to set that up so the Arduino can trigger that.

Thanks for any help.

You can use a reed relay or an opto FET across the switches
http://ca.mouser.com/ProductDetail/MEDER-electronic-Standex/SIL05-1A85-76L3K/?qs=sGAEpiMZZMv4tz1TW%2FArEwxS532n69wLIaTMPLw%252bZK0%3D

http://ca.mouser.com/ProductDetail/Fairchild-Semiconductor/H11F3M/?qs=sGAEpiMZZMugPZL2oX39yI7jG0lGkOJrZAn2RSkMB18%3D

Does it matter if the power source of the transmitter is different from the Arduino?

Unless you use a relay , you at least need to know the voltage (polarity,) across the switch in order to bias a transidtor or Fet correctly.
You need to measure the voltage across the swivh to determine which trrminal is positive anx which is negative. (how else can you
know which way to connect a Fet or transistor ?

It's measuring 3.3V at the switch.

Does it matter if the power source of the transmitter is different from the Arduino?

The two suggestions made will totally isolate the arduino from your unit.

Looks like the unit is powered by a battery.
Also the two switches are of low current and low voltage type.

My guess either the reed relay or the Opto FET will work fine.

It doesn't matter what the voltage is. All that matters is tha you mark the positive terminal + and the other one -.