While those independent prosecutors launched the investigations of Presidents Richard Nixon and Bill Clinton, there is no comparative figure in the House impeachment inquiry. Former special counsel Robert Mueller led the Russia probe, but no new prosecutor has been tapped by Attorney General William Barr for the Ukraine matter. That leaves House Democrats with only a whistleblower's complaint rather than boxes of investigators' evidence to guide them.
"Congress has to do that," Schiff said, because the Justice Department believes "there's nothing to see here."
Schiff, the chairman of the House intelligence committee, is leading the probe at the direction of House Speaker Nancy Pelosi and proceeding like the prosecutor he once was, staging a grand jury-like process that has been pilloried by Republicans. As Schiff works behind closed doors to build the case, Republicans accuse Democrats of waging an unfair — and according to the White House, illegitimate — investigation. But Schiff says the House has few other choices than to build the case on its own.
Working from the outside in to collect evidence, Schiff is probing Trump's attempt to pressure Ukraine into investigating Democrats in the 2016 election and the family of his 2020 rival, Joe Biden. That means private hearings, keeping the witnesses separated, and the depositions closed, to prevent people from coordinating their testimony or concealing the truth.
"The special counsels in the Nixon and Clinton impeachments conducted their investigations in private and we must initially do the same," Schiff wrote in a letter to colleagues this week.
Rep. Emanuel Cleaver, D-Mo., praised the work of House Democratic investigators but contrasted their resources with those of Mueller, who had a bigger budget and assistance from the FBI.
Cleaver said officials caught up in the Ukraine probe "ought to be thankful we have to use our own investigators. If we had the use of the FBI, things would be turning up daily. Something happens when the FBI shows up."
Republicans complain the investigation denies due process to the president and subpoena power for the minority party to call its own witnesses.
"Adam Schiff is not a prosecutor in this case," said Louisiana Rep. Steve Scalise, but he's "acting like one and doing it in secret without fair rules."
While the push-and-pull over process is typical for Congress, the stakes are anything but ordinary, part of a broader struggle over impeachment heading into the 2020 elections. The debate could help determine whether Democrats succeed in convincing voters that Trump's conduct was a gross abuse of power, worthy of removal from office, or Republicans help him beat the charges, both in a Senate trial and the jury box of public opinion.
Richard Ben-Veniste, a former Watergate prosecutor, said it's a "target rich environment for the investigation of misdeeds by the president." He expects the probe being conducted by the House committees will eventually result in articles of impeachment against Trump.
"It will then be for all of us at the end of the day to judge how well of a job they've done," he said.
Schiff says the transcripts will eventually be made public. For now, much of the action of impeachment remains behind closed doors.
Democrats have interviewed a series of witnesses from the State Department, and are now pivoting to the Pentagon and the White House budget office as they investigate why the Trump administration withheld military assistance from Ukraine over the summer.
Lawmakers are adapting to their role. Democratic Rep. Eleanor Holmes Norton, the delegate from Washington, D.C., said she has been listening intently to the testimony.
"I was looking for inconsistencies, and all I can tell you is, I have not found any," she said. "They're terribly credible."
Republicans say that if Democrats want to compare their inquiry to the legal process, the president should have counsel in the interviews and be able to cross examine witnesses. They say they should be able to call witnesses, as has happened in previous impeachment proceedings. They say the transcripts should be released so the public — and lawmakers not on the three committees leading the inquiry — can see what witnesses said before the House moves to an impeachment vote. And they argue that the full House should vote to formalize the inquiry, a move Pelosi has so far rejected.
"This process is a joke, and the consequences are huge," said Rep. Lee Zeldin, R-N.Y.
The GOP leader, Rep. Kevin McCarthy, said Republicans will force a vote Monday on a resolution to censure Schiff.
Some Democrats say they had initial concerns, as well. Rep. Elissa Slotkin, D-Mich., is among several freshman lawmakers from national security backgrounds stood up at this week's private meeting of House Democrats to question the process and ensure that Republicans had a fair chance to participate. She says she was assured by the answers.
"Indeed, the reason why these hearings are going on for eight hours and nine hours is because both sides ask questions until they're done," said Slotkin, a former CIA analyst.
As one of the newcomers politically at risk over impeachment, Slotkin said she was happy to take a vote to formalize the inquiry, but finds it unnecessary.
"I've already done three town halls on this, taken it on the chin, I just don't think that's the problem because the vast majority of us who could be the most politically vulnerable have already come out."
Rep. Seth Moulton, D-Mass., a veteran and member of the Armed Services Committee, said Republicans "are trying to talk about process because they don't want to talk about substance. They don't want to defend what he did."
"What we understand is that every detail that comes out makes the story worse," Moulton said. "What we already know is enough to proceed with impeachment."