What happens to the enzymes produced by the digestive system?
The answer is autodigestion fascilitated by denaturation of the digestive enzymes.
EVIDENCES OF AUTODIGESTION OF THE DIGESTIVE ENZYMES
1.The concentration of digestive enzymes is found to decrease from the small intestinal ileum to the large intestine.

There is a general trend to a decrease in the enzyme concentration the furthur down in the intestine the samples have been collected inspite of a concentration of the intestinal content in the same direction (see Figure1).This fact must indicate that the enzymes are partly inactivated or autodigested in the lumen of the small intestine.[pg 1529].

2.Essential amino acids have been found in the small intestinal lumen even if the food ingested didnot contain them which suggests that their source was the hydrolysed digestive enzymes which would eventually be re-absorbed in the small intestine to prevent their loss from the body.

Amino acids from dietary digestion are not alone, because the ingestion of food-even non-nitrogenous food-stimulates the digestive tract to secrete endogenous protein, derived from the sloughing of intestinal cells and used up digestive enzymes. These recycled proteins are a rich source of essential amino acids.Studies by Nasset show that regardless of the amino acid mix of the meal, the intestinal tract maintains a remarkably similar ratio of essential amino acids.[source]