Yes, they have more authority.
The Senate shall have the sole Power to try all Impeachments.
Some interpret this to mean the Senate can not only decide to call witnesses, but - in essence - also to decide if it's legal to do so:
The Senate (by majority vote) retains the right to decide
the privilege question either directly or on an appeal of the Chief Justice’s ruling. In casting their vote and
exercising their own independent constitutional judgment, individual Senators may consider important
questions such as whether executive privilege is applicable in an impeachment trial and, if so, how to
balance the Senate’s interest in obtaining information necessary to an impeachment trial with the
President’s qualified interest in confidentiality.
Congressional Research Service - Obtaining Witnesses In an Impeachment
Trial: Compulsion, Executive Privilege, and
the Courts
(pdf),
If this happens, courts may get to decide if the Senate does indeed have such power, and if they decide in favor of the Senate the question about executive privilege, which the House subpoenas were blocked by, becomes irrelevant.
This answer is not arguing that the Senate will or should do things that way, but that they could do things this way. Which provides a Senate trial with greater authority to call witnesses than a House impeachment hearing.
This article provides a more thorough, but also more partisan exploration of the question, by a leftist opinion writer.
A comment raised that this is equivalent to the House's sole Power of Impeachment. I disagree. It's a lot easier to legally argue having "the sole power to try Impeachments" nullifies other courts' input on which witnesses may be called during the trial, than it is to argue that the House having the "sole Power of Impeachment" doesn't allow courts to be involved in legal questions during the investigations that precede the House's vote on Impeachment.
Some go even further, and argue the word try in The Senate shall have the sole Power to try all Impeachments means that refusing to call witnesses would be unconstitutional:
Senators are thus constitutionally bound to follow what Justice White described as “a set of minimal procedures.” His opinion does not specify their exact contours, except to say that they must be the kinds of procedures a reasonable judge would regard as necessary components of a court trial. Because no reasonable judge would refuse to allow witnesses with personal knowledge of the facts to testify in an ordinary trial, it is the Constitution itself that establishes the right of House managers to call witnesses The Atlantic - An Impeachment Trial Without Witnesses Would Be Unconstitutional
In addition, a ruling by the Chief Justice of the United States John Roberts on the matter (while presiding over the impeachment trial) would be immediate, and carry far more moral authority than a party line vote in the House.
