Same ride, other direction
Son and I did the same ride, but going north in 2007. There IS a customs office on the left just as you crest the hill after leaving the center of Puerto Acosta (where the police stamped that you had passed that way). We did NO paperwork before leaving, and found NO CONTROL on the Peruvian side, so we illegally left and entered a country!
Not so much of a problem for us as the bikes are registered in Peru and we know the immigration people and they just stamped us in to the country a week later as if we had just arrived from Brazil!