Given what we are now learning about the diverse mix and variance of microbiomes (aka bacterias via human gut project), is it possible that synthesis of aminos can be done by specific bacteria and therefore provide the "essentials" if atomic ingredients are available in digestive tract?
If your body was filled with highly specialized bacteria living in nutrient-rich conditions, what you suggest could happen*.
But in practice, you're not filled with such bacteria, and most of the bacteria in your body are living in rather harsh conditions for rapid metabolism.
That said, gut bacteria do produce a range of other useful nutrients. Look for a review on gut microbiome metabolism, and you will find a number of examples.
*Some basic calculations: Say we take an optimistic estimate for productivity of 1g AA/g biomass/day. (It's what they got in this paper (one of many) that characterized a strain of Corynebacterium glutamicum engineered to produce lysine for industrial purposes and growing in a bioreactor. There are better strains now, but still, let's call that an unbelievably optimistic estimate of what a bacterium in the body might produce in the right conditions). How much bacterial biomass is in the body? Not a trivial question to answer (How much weight/volume do microbes occupy within the human body?) but let's go with the middle of the road estimate of 1kg. So 1 kg of awesome lysine-producing bacteria might potentially produce 1 kg of lysine a day. Googling around for dietary requirements seems to get you somewhere in the 1-5 g/day range. So assuming vaguely similar productivities for other amino acids, we've got plenty of room to spare if we assume that our bacteria are highly specialized for this function and are exposed to friendly, oxygen-rich and nutrient-rich conditions.