How can I measure the current draw of a device without powering it?
No, any static test could easily be quite a bit off.  Without actually powering it, you don't know what the current draw will be.  However, if this siren uses a motor, then then worst case current draw will be at startup.  That's the value you want to design to anyway.

Non-motor types could possibly not draw much of anything until they got to a particular threshold.  You really have to run it.  Put some muffling around it and run it for a second or so.

Also note that what you need to design the relay or transistor switch to is both the worst case current draw and the average current.  The parts needs to be rated for the worst case current, and not drop too much voltage during that time.  The thermal design has to deal with the heat from the average current draw accross your switch.

All that said, its hard to imagine it will be more than a few amps at 12V.  Get a 10A relay and be done with it.