my understanding is that AMCAS makes the letters you designate to each school available to that school, and they then download it when they feel like--though generally quickly.
I have waited so freaking long for my committee letter that as soon as I get notification that it's on AMCAS, I am calling each of my schools to 'check if i'm complete' aka get them to look for my letter. Overkill maybe, but I'm extremely impatient and frustrated with my committee right now.