Falsehoods associated with the interplay of Roman/Italic folk traditions and those of the Greek homelands are some of the most common modern misconceptions of the ancient world. The idea that the Romans merely adopted Greek mythology for their own is simply wrong. Everyone has folklore, and the Romans were not a barren, blank slate in this regard, waiting to be "imprinted upon" by someone else.
Because many European folk traditions in the ancient worlds drew on a common Indo-European legacy of story and beliefs, there were similarities shared by many people. At the same time, because folklore is constantly in flux, the various European traditions included variations and at times contradictions. The mere act of ancient authors - whether Greek, Roman or Egyptian for that matter - synthesizing a variety of stories into a single text - was a process that likely did some violence to local traditions as they existed with all their contradictions among the "folk." Greek stories and beliefs on one island or in one valley were not likely to be identical to those of another. By writing down an "official version," creating what we now refer to as "Greek mythology," authors were creating an artificial synthesis that more or less (often, likely, less) reflected what was told and believed on the local level. Nevertheless, the written word has always been a powerful thing, and once codified in writing, a text could influence folklore on the local level.
As Roman influence expanded, Romans encountered beliefs that often seemed very similar to their own. Much of the cultural history of the Roman empire consists of reconciling their traditional beliefs and stories with what they encountered in various places. This resulted in many hyphenated local gods as Romans recognized similarities and yet tried to reconcile themselves with diversity within their empire. On some level, the Romans were acting like early folklorists, collecting local stories and then dealing with similarities that they encountered. But that's getting ahead of ourselves.
Early on, early Romans encountered Greek colonies and recognized the extremely impressive facts that 1. Greek mythology seemed oddly familiar (even when it was also different from their own) and 2. the Greeks had all of this documented in some remarkable forms of literature. Very impressive!!! As a consequent of this, the Romans began the process of attempting to reconcile their own diverse traditions with those documented in writing by the Greeks.
Because of a common Indo-European legacy, many of these stories and gods were akin to one another - even when there were differences. Dealing with those similarities and differences was extremely influential as the Romans codified their own folklore into an established, written mythology.
The Roman, then, did NOT go shopping for a mythology and decided to purchase the Greek version, taking it off the shelf and adopting it as their own. Instead, Romans encountered oddly similar stories that were impressively documented in works of great literature. This encounter was influential as the Romans were codifying their own stories and beliefs in writing, but we are not looking at the adoption of foreign myths; we are looking at the influence of Greek literature on Roman traditions.
This is not an unusual process: all people have folklore, and folklore by its nature diffuses and changes. People have their own folklore even as they encounter other people stories; other people's stories are impressive, so people adopt and adapt them as their own. This is why we find similar stories scattered among diverse people (which can be attributed either to descent from a common ancestor or diffusion from a single place). This is not because one group of people had a great body of stories and their neighbors had none. It is because people are always in the market for a great story and people eagerly borrowed and adapt material to fit their own traditions. Folklore is an organic, living process, and that is how Romans dealt with their many encounters with other people and their traditions.