In a person without diabetes, a rise in blood amino acid concentration (the result of protein metabolism) stimulates the secretion of both glucagon and insulin, so their blood sugar remains stable. But in people with diabetes, the release of glucagon without insulin or with impaired insulin response can cause our blood sugar to rise precipitously several hours after a meal high in protein.
The insulin is secreted to stimulate protein synthesis--the uptake of amino acids into muscle cells--making them less available for gluconeogenesis. The glucagon is secreted to stimulate the uptake of amino acids into the cells of the liver for gluconeogenesis.
So why are these two hormones battling for opposing uses of the same amino acids? Isn't that non-productive?
Actually, the phenomenon serves an important purpose. As you probably know, insulin lowers the blood sugar, while glucagon raises it. In the non-diabetic state, the release of these two opposing hormones ensures that the amino acids are used for protein synthesis (because of the extra insulin) but the blood sugar doesn't drop to dangerously low levels, even if the meal was low in carbohydrate. As a result, blood glucose concentration remains reasonably stable during protein metabolism. The insulin and glucagon essentially cancel each other out in terms of their effect on blood glucose, while the insulin is still able to promote protein synthesis.
But in people with diabetes, as I mentioned earlier, the release of glucagon without insulin or with impaired insulin response can cause our blood sugar to rise precipitously several hours after a meal high in protein. This is due not only to the glucagon's directly raising the blood sugar, but also to the fact that in the absence of insulin it increases the amount of the amino acids that are used for gluconeogenesis.
For what it's worth, arginine is the amino acid reputed to stimulate the highest production of both insulin and glucagon.