While communication between the hemispheres and just separate lobes and faculties in general could matter a lot, I don't think the left vs right hemisphere thing should be overstated or polarized. The thing about the corpus callosum and about the functionalities serving as filters as switchboards is definitely important but doesn't necessarily imply such an emphasis on left vs. right dichotomy.
Anyway I wouldn't look to psychedelics for improving ability to analyze patterns and stuff like that, but if anything to encourage making more connections in your brain and thinking including less obvious ones. Of course there are problems associated with "too many connections or too much pattern resolution", also associated with e.g. schizophrenia and autism but the opposite with depression among other things.. although I am definitely not saying you can just acquire those but simply that it is not without risk, there are other manifestations.
Getting more connections in your mind, especially unlikely ones... goes hand in hand with the natural pruning that occurs. It's never 'the more connections the better', but if you have more flexibility in connectivity you get different pruning and a different end result.
I would guess that psychedelics may help you to think outside the box due to the more/unlikely connections and there may be a few questions on an IQ test where that could make a difference but that is something different than somehow becoming a person with more analytical skills overall which seems to be more about your style of thinking and genetics.
Autism by the way appears to have something to do with layering of neuron surface cells leading to some regions with anomalies where synapses may better form. This may lead to "shortcircuits" but also just a different organization of connections and more unusual ones which is perhaps good for the abstract but not so much to keep things unambiguous. If these effects are too extreme and the processing of the brain cannot cope then you might imagine the more classic dysfunctional cases.
Of course there are a lot of other phenomena associated with autism, but it is always most interesting to see which phenomena cause a lot of others top-down.
I think there is a good explanation in all of this for my propensity for tripping, it seems to be in line.
Both nootropics and psychedelics (microdoses are popular now) are considered interesting ways to enhance your mind. But I think it is not about the ability to improve your brain's horsepower per se but just the connectedness and integration which by the way are also keys to creativity and emotional equilibrium or balance and harmony in your mind. There are probably more signs that I could list and they also are closely related to depression which often comes with the opposite: being more stuck and isolated in your head and instead of seeing the possibilities running in (panicked) circles which kinda get engrained.
P.S. based on a test many years ago i got into mensa but I never really got much out of that cause of social reasons (and more recently the fact that they basically have the worst service ever from an experience i had) - anyway much later I underwent some different kinds of tests which also included some IQ testing but I was using so much MXE during that time that it is not surprising that I scored lower. Not bragging, I am ashamed of that if anything - and its bullshit anyway to take credit for something you didnt personally accomplish.
For me racetams are lovely but I am on a dexamph script now and racetams make more irritable and borderline insomniac anyway. But I think i might use it again if I feel that my mind needs to stay creative and flexible. Never let it atrophy!