Wouldn’t one try to eliminate the other — as humans supposedly eliminated the Neanderthals? (Or others — as in NicknamedBob’s #47)
Not necessarily. One problem regarding the Neanderthals is that they supposedly occupied the same ecological niche.
Perceive the world today. We are "partnered up" with any number of species. Elephants, horses, cormorants, canines of all varieties, even hawks and selected species of rodents. Many of these are "working relationships" in which each species gains something from the relationship. Even common domestication can be viewed that way.
But this does not address sentience. It merely emphasises that dissimilar species can get along, by each working in a specific ecological niche.
Some who work closely with very intelligent animals, even including some of our pets, will aver that the animals often exhibit every characteristic of sentience. This is not my argument. But evidence can be gathered that we have it in us to cooperate with other species.
Let us consider a hypothetical. Suppose we finally crack the language barrier that separates us from "dolphins", the cetacean variety of mammal so comfortably at home in the sea.
It is clear that a form of trade could be developed, from providing safe shelter for the raising of young, to exotic forms of communication such creatures could use. We would benefit from their ability to navigate the oceans in three dimensions, perhaps herding fish for our mutual species to harvest.
It can be seen that this would be an easy and comfortable arrangement, with neither species having either the capability or the inclination to harm the other. Primarily this is a result of our ecological niches not being in direct conflict.
I really don't see a problem, and we should perhaps hope that it is so, for eventually we may encounter another species just a little too reminiscent of our own former bloodthirsty selves.