To begin with, it must be recognized that all legal travellers to-and-from nations of intelligence interest are subjects of inquiry by US intelligence services, principally the CIA and FBI. However, other instruments of government security such as DEA and the various military intelligence agencies, as well as other governmental services that have international briefs, are also authorized to gather appropriate, and ofttimes classified, information on activities of foreign and domestic travellers. In fact, depending on interest in and personal ability to proceed, all major nations, and to some degree and interest, even minor nations pursue the same activity.
In this regard, travel to and business negotiations with Russia by Donald Trump and his various representatives would naturally be a matter of interest for US embassy personnel assigned to track such contacts. Most businesses make a point of touching base with the State Department and the in-country embassies alerting the official instruments to their intentions. The Trump organization followed these usual procedures. A problem arose in Moscow when the extent of Donald Trump and his representatives' contact with Russian political and business personalities became infected by inadvertent involvement with Russian undercover Intelligence officers in the course of regular business. From the Russian standpoint, Trump was a target of opportunity as a successful multi-billionaire American businessman with reported good political connections within the American and other nations' political environment. And from that standpoint everyone was just doing their job.
When Mr. Trump became a candidate for president in 2016, the Russian intelligence services and the American services had to review the information that they had on him, as did nations such as China and others more benign. The review became of more immediate importance when Christopher Steele, a retired officer in the UK Secret Intelligence Service, generally known by its WW2 name as MI6, was hired to revise and enhance an anti-Trump dossier that was in preparation by an organization financially supported by the Democratic Party.
Steele had established what he called a "commercial intelligence firm" named "Orbis". It is unclear whether Orbis truly was a legitimate private firm or a cover company for MI6 activities. In any case, as a retired officer, Steele would have had to check back with his former service to gain approval to proceed on a project that would involve activity that might influence the U.S. election. MI6 would have instructed Steele to clear and coordinate the matter with the FBI as it was a US domestic security issue. MI6 liaison with the CIA would have had to be informed as well as The Security Service (MI5), the UK domestic intelligence agency that is the actual counterpart to the FBI. In other words a round-robin of coordination would have occurred
As the FBI already had information on Donald Trump as a legal traveler to Russia, and therefore an existing interest in him, they would have been desirous of having access to anything Steele evolved. To be clear, this would have been the same case no matter which American billionaire/internationalist would have been involved. Most of these high profile personalities or their aides are interviewed pro forma regarding anything with which they may have observed of unusual interest during their trips to countries of strategic importance. And to the extent manpower allows, so would have other less public businessmen. There is no requirement that these travellers must agree to be interviewed. It's all on a voluntary basis.
Steele, therefore, would have had to agree to pass on his findings to the FBI who, in turn, would be expected to pass the "take" on to the CIA as the official liaison with any MI6-connected activity. Whether Steele was paid by the FBI or not is irrelevant. By sharing information on whatever he gathered and/ or created relative to Trump, Steele was acting with the Bureau's knowledge and thus to some degree, effective approval. It could be argued that Steele in some manner was under the operational guidance, even if not under control, of the FBI. Whatever the legal niceties, the fact of the entire Steele operation had to be tightly held at the highest levels of the FBI.
Meanwhile, an additional individual may have been added to the Steele project in the form of another former MI6 officer named Pablo Miller. He had been a compatriot of Steele when the latter served in the UK Moscow Embassy. Highly decorated, Miller had been substantially involved in the defection of Sergei Skripal when the latter was a GRU officer and well known in intelligence circles. Skripal was arrested by the GRU and years later exchanged in a spy swap with the UK. Information suggested by former UK Ambassador Craig Murray indicates that Miller may have been useful to Steele in the latter's assignment to gain negative information on Donald Trump from Russian sources. (Unrelated, but interesting, is the coincidence that Miller was a resident of Salisbury where the exchange prisoner Skripal now lived.) This circumstance suggests Miller may have been still on MI6 payroll acting as a "minder" of Skripal before the latter was poisoned and while Miller was aiding Steele on the side. It may seem complicated to the uninitiated, but not at all in the world of intelligence.
The collection operation for material now known as the Steele dossier apparently became too sensitive for the FBI case agents, and their relationship with Steele was terminated. The decision to cease participation in the Steele project by the FBI was made after portions of the dossier development were leaked to the press. Whether the latter situation was resolved in consequence of Steele's own action or the sponsoring organization, Fusion GPS, is unclear. In any case, Steele and his coordination with high levels of the FBI was closed down. The possibility of a compromise of the FBI-Steele relationship had become politically too hot to continue with the relationship.
Peter Strzok, as Deputy Chief of counterintelligence for the FBI, would have been fully knowledgeable of the Steele relationship and the latter's activity on the part of Fusion GPS. In fact, Strzok may have acted as the Bureau's case agent on this project. It is his knowledge of the operation in all its complexity that keeps him still employed there completely divorced from any operational responsibility in the FBI. Clearly the Bureau finds this method of retaining Strzok as an employee holds greater leverage legally and security-wise.
For his part, Steele and his family - as well as Miller - live under the protection of MI6 and appropriate authorities in undisclosed facilities. Steele has surfaced only to respond to a defamation suit in a London court brought by a Russian businessman who claims he was unfairly characterized by Steele in the famous dossier. The Russian says that he intends to proceed further in the U.S. with his suit. Meanwhile, the domestic political scene in the U.S. is littered with people on both sides of the last election who have been or are being brought to task over their involvement - rightly or wrongly. There are no signs that Pablo Miller's purported involvement in gathering information for his friend Steele to use in the infamous dossier had anything to do with the poisoning of Skripal and his daughter. The problem is that as one digs around, the ball of twine unravels only to reveal another ball of twine. The funny thing is that modern-day politicians seem to thrive on each new disclosure and/or promise of disclosure. One wonders how history will deal with these complications. But then there'll just be more complications and more balls of twine.