Not really. They simply get to participate if they are
named to a committee by the person assigned to lead the
committee, or the Senate Majority leader.
The Majority leader is in control of those things, I
believe.
From: “A Survey of House and Senate Committee Rules on Subpoenas”
Updated November 12, 2021
In the summary, “Most House committees have also delegated to their chair the power to authorize subpoenas.”
https://sgp.fas.org/crs/misc/R44247.pdf
So, the protocol is that only the House committee chairs have subpoena power. A huge perk to being in the majority.