We do know these two facts:
- There is (horizontal) gene transfer between the host and the Alien, as the Alien does assimilate some of the host's physical attributes (as seen in Alien3 and the Alien vs. Predator franchise especially). See this well-sourced article on the AvP wiki for more information.
- Aliens have genetic memory, ie. memory that is inherited from one generation to the next (as seen in Alien Resurrection and detailed in several novels).
Of course, humans don't have genetic memory in the first place, so the gene transfer from a human host (or a canine/bovine host, as in Alien3) to the chestburster Alien will probably not transfer any memories.
And we don't ever see evidence that a regular Alien assimilates memories from a regular human host, to my knowledge, although it might manifest itself as just regular intelligence and instincts that are hard to differentiate from the Alien's. This is theorized in the book Alien: The Weyland-Yutani Report (apparently on page 25, although I don't have it in front of me now).
On the other hand, I don't see any reason why Aliens couldn't assimilate the proper memories of a host species that does have genetic memory, although I'm not aware of any instances of this from canon.
In Alien Resurrection we see something similar, though. When Ellen Ripley is cloned from a sample of her blood, taken when she was a host for an Alien queen, her blood had already fused with the Alien chestburster. The result (after some trial and error) was a Ripley impregnated with the chestburster, and the two shared genetic memory. This is seen in the movie in several instances, like Ripley retaining her memories from before her death and a kind of shared emphatic "hive mind" with Ripley and the Aliens, especially the Queen and the "Newborn".
The answer doesn't really change depending on the movie. We didn't really see the gene transfer in action until Alien3, where we encountered the first Alien not born from a human host, but according to the documentary Alien Evolution (and other sources) it was always planned as a feature of the Alien, and the Alien we see in the first two movies is just the human version of it. The genetic memory didn't appear until the fourth film Alien Resurrection.