In the absence of a signal, or in the presence of a constant signal, the automatic gain adjustment on the receiver will either lower or increase the threshold until it gets a duty cycle of around 50% (ie, it ramps the gain up or down until finds a duty-cycle around 50%, even if that means it's outputting noise). This is why you see the same thing with transmitter on as with it off - it adjusts the gain until they're the same.
(also, sometimes those XD-RF type receivers just plain decide to suck - I've seen them get max range of ~10 centimeters under conditions I don't really understand)
If you use receivers based on the SYN470 chip, you can tie the CTH pin to Vcc through a 5 mega-ohm resistor to keep it from cranking the gain up so high that you get random noise when the transmitter is off. Remember however, that you can still get interference from other 433mhz devices like garage door openers, car door remotes, doorbells, and other common household items.
The right way to use these transmitters is to send a pattern with them that the microcontroller on the receiving side can verify. This could be done either through the VW/RH libraries, or you can roll your own (this might be useful - this is my homebrew protocol - though I don't claim it to be superior to VW/RH by any means, and there are many obvious improvements) https://github.com/SpenceKonde/AzzyProjects/blob/master/433mhz/txrxtoserial21/txrxtoserial21.ino ). Typically you send the signal a few times in succession, often with a "training burst" of 50% duty cycle preceeding it to try to get the AGC on the receiver to adjust to the desired gain. The SYN470 boards in general work much better than the XD-RF ones (which are a simple superregnerative receiver, while the SYN470 is superhet).
Per the datasheet of the SYN470, the highest demodulator bandwidth available is 10kHz. That implies that the shortest response time you could hope for is ~100us - however, even that is not attainable in light of what I discussed above.