Can you give us more info. Or maybe rephrase the question. Your current question gives the impression that your toilet rattles constantly until you flush it? If this is indeed the case, I believe you will need to clean or replace the fill valve in the toilet, that should solve your problem.

I agree, we could use more details. I don't think you mean the toilet, china bowl, actually rattles but maybe the fill valve or supply pipe.

The rattling starts up randomly and won't stop until I flush the downstairs toilet.

What is rattling: pipes, toilet, fill valve, etc...

The rattling starts up randomly and won't stop until I flush the downstairs toilet.
Describe the noise, Is it a tapping sound? How loud is it? Does it start up when any other fixture is turned on? Where's the noise the loudest? Have you ever pulled the lid on your toilets and hecked the water level in the tanks? The water level wants to be 1/2' under the white overflow tube. Check to make sure a ballcock isn't kicking on intermittently. Could it be a flapper's leaking a bit causing a ballcock to run.? Check all those and get back to me. Regards, Tom