Creating a wireless transmitter/receiver for off-the-shelf siren
Radio isn't that simple. Imagine your siren built into a box with a radio receiver (just one of those cheap ebay radios that cost a few dollars). The radio will be constantly producing a noisy data output because it is a high-gain system and is designed to try and extract pico watts of power from the "ether" so that a weak radio broadcast can be adequately received and decoded. 

A slightly more expensive receiver may have an RSSI pin - this will give you some Indication about the level of the Received Signal Strength. You could attached an op-amp or comparator so that you gate the noisy data output to your siren only when the signal strength is fairly high. This would help you exclude all the mushy noise that your receiver would naturally pick up in the absense of a strong signal.

However, the strong signal you receive may still be from other equipment so your siren will falsely go-off quite often.

What proper EEs do is transmit a message that is equivalent to "on" and this is usually formed from: -


A preamble of several bytes that allows the receiver to adjust its receiver in time for the next part of the message
An address byte so that there is a nod towards trying to make the "system" have some level of integrity/security
A bit that might mean "on" or "off" (your payload data).
A two byte error check so that even more integrity is afforded.
Maybe send it twice and, only act on two consecutive transmissions being correctly received.


This is a minimum system and won't guarantee against false alarms every week or so but will be reasonably reliable. You may think that this answer is aimed at those folk wanting to send data messages as per this: -


  but most people ask for more complicated things like transmitting more
  than on/off


I'm sorry to disappoint you but sending one bit of information (i.e. on or off) requires signal integrity to be achieved by transmitting a message as per the bullet points above.

Just think about a music radio receiver tuned to an unused part of the spectrum - all you hear is noise and that noise is converted to thousands of ones and zeros in order to extract data - the whole point about a data transmission with integrity, is that that chances of noise exactly matching the random ones and zeros produced by a good transmission is quite slim so, you package your single bit (on or off) into a frame that has high integrity. This means that the receiver can detect good transmissions relatively easily AND bad "transmissions" (due to noise) only correspond with the frame protocal once in a while.

Transmitting 1 or 0 by radio isn't as simple as you think (and for good reasons).