Continuity isn't exactly Red Dwarf's strong point, so it's really difficult to say, since Red Dwarf contradicts itself a lot - often simply for jokes. The spare heads appear in three different episodes, and are referenced in a fourth, as a one-line joke.
In DNA, Kryten has been turned into a human, and goes to talk to his spare heads. His spare heads refer to him as "you", and they have an argument. But it is unclear whether this is similar to the permanent situation, or if the situation was changed by Kryten's temporary humanness. In the same scene, Spare Head 3 is shown speaking in a yorkshire accent and displaying a completely different personality. This is explained by one of the "normal" heads as him having "droid rot". It clearly affects his language and personality, but it's unclear to what extent this reflects the normal interaction between the spare heads.
The next time the spare heads appear is in Tikka to Ride, the episode you reference. At one point, after refusing Lister's idea for a trip back in time, Kryten says he wishes to "off-line for the next twelve hours while I discard some old cache files". He then explains that this involves plugging his head into the ship's computer for twelve hours, which gives Lister the idea of using a spare head. Lister plugs the spare head into Kryten's body, where it exclaims surprise at being "head head", refers to Kryten as "he", and suggests that Lister disables his guilt chip, to enable him to agree to the curry trip. This interaction certainly suggests that the heads do to some extend have a separate identity. Later in the episode, Lister says "I nicked Kryten's body. That's spare head 2" which further supports this theory.
In Stoke Me a Clipper, Kryten makes reference to the fact that his other heads voted him the "big-eared, ugly one".
The final episode in which Kryten's spare heads appear is in Beyond A Joke. In this episode, Kryten's head explodes when Lister asks for ketchup with his lobster. As his spare heads are attached and activated, they each explode in exactly the same circumstances. Later in the episode it is discovered that the reason Kryten's heads were exploding was because of a build-up of emotions in Kryten's "negadrive". While it's not explicitly mentioned, it would seem that this "negadrive" is located centrally, on his body somewhere, and that this causes his heads to explode with little to no control from the heads themselves.
From all the (rather spotty) evidence, I would conclude that yes, Kryten's heads have a distinct identity, and can develop separate personalities; though it's entirely possible in Red Dwarf for the above examples and situations to be nullified by a joke in a future episode, since the show doesn't put much faith in preserving continuity. In addition, it's highly likely that the above aspects of Kryten are written solely with the humour and stories in mind, rather than with any central idea of the scientific/robotic side of how Kryten works.