Rikudo gave Sasuke and Naruto a temporal part of his chakra and also Kaguya-sealing techniques.
That chakra made Sasuke awake Rinnegan and have black chidori, and made Naruto become NaruJesus.
Only Naruto had Nature chakra, but because both received Rikudo chakra, they both were able to hurt Juubi Junchurikis with it (Naruto was already able thanks to Nature chakra, though, so it only meant a real upgrade for Sasuke).
The proof that Sasuke can hurt Juubi Jinchurikis thanks to Rikudo chakra is his black chidori. When Obito gave Kakashi the double mankegyo, Obito had already become a Juubi Jin himself, therefore, he had tasted the power of Rikudo which he temporarily passed on to Kakashi after dying. This power also allowed Kakashi to hurt Kaguya, another Juubi Jin, thanks to his own black chidori, just like Sasuke.
After Naruto vs Sasuke's fight, their Rikudo powers vanished, but Sasuke managed to keep an already awoken Rinnegan. Naruto kept nothing, aside from the scattered pieces of chakra from all the bijous to turn Naruto into a Bijou Hub.