To make this argument is balanced we have to assume that neither users knew what the other one's ability did, we also have to assume that they've both achieved all there is to achieve in their respective skills and they are of similar ninja skill level. As far as we know the Rinnegan has the ability to control 6 bodies simultaneously while being able to cast any jutsu of any element. Also it has a wide range of summons, a body that absorbs all damage and ability to see through all the eyes of the collective 6. Sharingan on the other hand has advanced vision allowing the user to track movements precisely and copy them, also allowing to copy certain jutsus provided that the user has the skill to cast them with their own abilities. Also it has the MS which allows 2 advanced jutsu to be used, one is a powerful genjutsu and the other is a powerful flame.
Lets not assume random things such as "Sharingan might have a hidden ability that allows the user to blow up anyone they see." Right now we are arguing which one would win with what ability we have seen of them so far. Even if the full range of Rinnegan's abilities haven't been shown I still think that the Rinnegan would win. Hypothetical scenario:
Sharingan user fights with Rinnegan user on even grounds ad they are both equally skilled, Rinnegan user is able to take the upper hand as the use has a wider range of jutsu and all 6 chakra elements. Sharingan user must fight back by activating Ms and catching the Rinnegan user in a world of torture practically rendering the Rinnegan user useless.
So far the Rinnegan user has learned of the Sharingan's advanced genjutsu ability, advanced sight and maybe ability to copy jutsu if the Sharingan use decided to use it during the fight.
The Sharingan user has learned that the Rinnegan has all chakra elements and is able to cast virtually any jutsu, and maybe that it also has reasonably advanced sight. But when it catches the Rinnegan user in its genjutsu it thinks that it has won.
The Rinnegan user sends out 2 bodies that are well suited to fighting against the Sharingan user and breaks the other one out of it's genjutsu, the zero blind spot and 3 vs 1 soon lands the Rinnegan in a higher spot, once the Sharingan user is no longer able to keep up it decides to use it's Black fire and burns the 3 bodies. By now the Sharingan user is exhausted and the Rinnegan user simply sends out 3 more bodies, revives the other 3 and claims victory.