No, the Sharingan users do not experience the full potential of the dojutsu. If that were so, the tomoe system would not be sensible.
From the Wikia article on Sharingan:
When first awakened each Sharingan usually will have only one tomoe (巴), although in Obito Uchiha's case, as well as Itachi and Indra's in the anime, they immediately gained two tomoe in each of their eyes. Through training and continued use the Sharingan will develop a second tomoe and then, upon full maturation, a third. When Hagoromo awakened Sharingan in the anime, he had all three tomoe immediately. All the Sharingan's abilities are available to the user from its earliest stage, but with greater development comes greater proficiency with those abilities.
So then, why different tomoe in each eye?
Since not much is being disclosed in the series about this, we could conclude an explanation from the given facts.
Sharingan broadly has two main abilities: the Eye of Insight and the Eye of Hypnotism. The evolution of the tomoe in each eye could mean the mastery of the respective abilities.