Relative measures are most relevant when you want to transfer your finding to other populations. Absolute measures are very dependent on the baseline incidence in the population under study, and will always be low if you are studying a healthy population or a low-risk group. That risk estimate would not transfer to a high-risk population. Absolute risk estimates are most relevant when communicating the results to a specific population.
Often one gets accused of reporting relative risk estimates, as they seem more dramatic. A 50% increased risk sounds more scary compared to an absolute risk increase of 0.5%-points from a baseline risk of 1%. However, I think the same argument can be applied the other way around, and those reporting absolute risks may be accused of doing so because they want to downplay, say, a 50% increased risk by pointing out that it only corresponds to a 0.5%-points increase. I think both are very bad arguments.
My opinion is that when reporting results from one population that is meant to be generalized, relative measures makes the most sense. This can always be translated to absolute risks customized to the population of interest when knowing the baseline risk of said population. So I guess always report relative measures, and if relevant also report what this corresponds to in absolute terms. At least when talking about risk, I can’t see a situation where the absolute risk estimates alone is a good solution. But this is my personal opinion only, and I do not have any specific sources in mind. I will follow this thread with great interest!