In Earthly ecologies, ecological niches are fiercely contested. In areas where wolves have been re introduced, the wolves have driven off the coyotes who had occupied the vacant niche. Similar things can be seen with the introduction of invasive species, if the "new" species has some competitive advantage over the existing species, they will overrun the ecological niche. This even happens with vegetation, such as the Kudzu vine.
In our own history, the Ancestors displaced the Neanderthals and Denisovians, and possibly other hominid species, most likely because the Ancestors moved in larger groups and were wired to cooperate more successfully than their cousins. This allowed them to exploit the resources of the land more fully, and living in larger clans and tribes the Ancestors had larger pools of skills and genetics - tiny groups of Neanderthals might fail if their one healer or tool maker died and could not be replaced in time, or diseases caused by inbreeding crippled the overall viability of the group. One might wonder if Neanderthal and Denisovian clans welcomed the Ancestors because they created a larger pool of potential mates (We know the Ancestors seemed partial to redheads, since the ancestral gene comes from the Neanderthals).
So multiple hominid species will eventually be reduced down to one. The only way multiple intelligent species could co exist (which seems implied with the question) is if they existed in entirely different and non overlapping niches, somewhat like humans and orcas. In that case, the interaction would be much different - if they were at similar levels of development then at the interface between the niches there would be stories of the strange and magical creatures of "the other place" who seem to be able to talk, and might be enticed to exchange gifts.
The situation where multiple Hominid species co exist only lasts for a short time, ecologically speaking. In the case of the Ancestors, they coexisted with the Neanderthals for several tens of thousands of years (likely because the Ice Ages limited the mobility of the various species), but once conditions had changed to the point the Ancestors began their expansion (eventually walking around the world), they rapidly displaced their cousins. If this was possible with primitive stone and wooden tools available to everyone, then the situation would be even more extreme if any species had more advanced cultures or technologies.