I've wondered that particular point. To what extent was the Mosaic Law restricted by secular law? I've wondered that in the case of homosexual acts. In Romans, Paul describes these as being worthy of death, but doesn't seem to specify whether the death penalty were to be carried out in the Roman occupation or secular society generally.
Yes only Rome could give the punishment of death.
Paul mentioned a lot more than homosexual acts.
Romans 1:26-32
26 For this reason God gave them up to vile passions. For even their women exchanged the natural use for what is against nature.
27 Likewise also the men, leaving the natural use of the woman, burned in their lust for one another, men with men committing what is shameful, and receiving in themselves the penalty of their error which was due.
28 And even as they did not like to retain God in their knowledge, God gave them over to a debased mind, to do those things which are not fitting;
29 being filled with all unrighteousness, sexual immorality, wickedness, covetousness, maliciousness; full of envy, murder, strife, deceit, evil-mindedness; they are whisperers, 30 backbiters, haters of God, violent, proud, boasters, inventors of evil things, disobedient to parents,
31 undiscerning, untrustworthy, unloving, unforgiving, unmerciful;
32 who, knowing the righteous judgment of God, that those who practice such things are deserving of death, not only do the same but also approve of those who practice them.
Although I'm don't think homosexuality was a crime in Rome.
I think the death here is the what will happen to those who practice such things when judged by God.