For of this you can be sure: No immoral, impure or greedy person—such a man is an idolater—has any inheritance in the kingdom of Christ and of God.
There are not two kingdoms, but one, that is certain. But the wording of modern translators sounds a lot like co-ownership. Read in light of the wonderful perspective and language of Ephesians, where Christ has received all authority from his God and Father, that co-ownership still strikes me as a good exegetical perspective.
However, there is more to be said. Granville Sharp, himself a Trinitarian, researched and published back in the early 19th century into Greek usages to promote Trinitarian claims based on Greek grammar. The title of his work is: "Remarks on the uses of the definite article in the Greek text of the New Testament, containing many new proofs of the divinity of Christ from passages which are wrongly translated in the Common English Version". What he establishes there is still frequently mentioned today (all modern translators will have heard of him and his rule). His rule stated that if two nouns of the same case (Greek has four cases, Nominative, Genitive, Dative and Accusative) are separated by kai ("and"), and the first of these two nouns is preceded by the definite article, then both nouns refer to one and the same subject. In his view, therefore, this removed any ambiguity regarding Titus 2:13 regarding the "and": Christ is the great God and Saviour!
"in the kingdom of Christ our God".
Daniel Wallace, probably alongside Bart Ehrman, is the greatest textual scholar around; he is also a believer who is also very careful about making unwarranted assumptions. I trust him. He found a number of limitations to Granville Sharp's model, which he had to tone down to allow for some inconsistencies. I have not read it yet (when I do I will update the blog with some highlights), but it will surely comment on this verse and why such a rule cannot be applied so haphazardly, thus also obliterating the Father from the picture in a headlong charge to prove the Son's divinity.
Attempting to prove the son's divinity is a dangerous task, and you have to be sure that you really are doing no harm to the other belief-sets into which you are tied as a Trinitarian tight-rope walker. It also can back-fire on useful research in other areas and on other texts like Titus 2:13.