Quoted:
The Manhattan Project was riddled with Soviet agents and Communist sympathizers from the beginning.
I wouldn't say that is accurate. Of the many thousands of people working in the Manhattan project, there were very few leaks; the primary spy was Klaus Fuchs. Greenglass was a technician who passed some limited information as well. It's believed that there was one other well placed source besides Fuchs - Morris Cohen's source, also identified by a code name in decoded Soviet cable traffic - but if they were ever identified it was not made public. It's also generally hypothesized that there may have been a leak from Hanford, because the Soviets learned a specific piece of information from the reactor operations there, but it's possible that Fuchs may have learned of that info and passed it along. Allan Nunn May provided some material samples and information on reactor design and some technical information but he was not at Los Alamos and didn't have access to actual weapon design information. The few leaks though provided very useful technical information to the Soviets.
There was a huge controversy at the time in the 50's were it was alleged that Robert Oppenheimer was a spy, personally I think it's complete hogwash. Oppenheimer tried to use his scientific/technical understanding of the issues to influence the government commissions he was on, which was the whole reason he was appointed to those commissions; when his opinions were in opposition to prevailing political sentiment he was thought to be a spy, rather than just having an honest disagreement based on his scientific knowledge. His personality was such that he often seemed very abrasive to those of lesser intellectual prowess, and I'm certain that this quality did not sit well with the political types that he was thrown in with on the commissions. There was information of critical importance to the Soviets that he was well aware of during the post-WWII and into the early 50's era that the Soviets were NOT aware of.
There was also a tremendous amount of information as well as material and equipment that was simply sent to the Soviets during WWII because they requested it through the lend lease program, and we gave it to them. For example, graphite of extremely high purity was required for the early design nuclear reactors, a level of purity that the Soviets couldn't duplicate. So they requested the graphite from the US and we provided it to them... the first Soviet reactor was made mostly with US provided graphite. There are many reports of massive amounts of technical printed material that were shipped to the Soviets via lend lease as well. Because of the secrecy of the Manhattan project, no one outside the project would have recognized or known the significance of the information that was provided to them. Information from Fuchs would lead them to simply request information on various industrial processes or technology, and the US would give it to them.
The US issued a somewhat comprehensive report immediately after WWII
(the "Smyth Report") to present some basic information about nuclear technology; this somewhat hastily prepared and not properly edited report inadvertently revealed a significant detail about reactor operation for plutonium production that saved the Soviets a huge amount of time by their not having to discover and overcome the same unforseen physical process that we did. It was only an accident of design that the Hanford production reactors were able to be made productive; in theory that piece of information alone might have saved the Soviets a year toward their own atomic bomb.
The Soviets had some reasonably competent scientists, but it truly saved them an immense amount of work with the information that they collected from espionage. Things like the chemical processes to separate plutonium as well as plutonium and uranium metallurgy which was learned through tedious and exhaustive research in the Manhattan project, and the Soviets didn't need to repeat. Most critical factors they ran experiments to confirm the espionage provided info, but it's a whole lot easier to just confirm an answer than to try to find it experimentally. "Joe 1", the first Soviet bomb, was an exact copy of the "Fat Man" bomb that was used on Nagasaki.
Whoever all the leaks were, it's virtually assured that there were none (from Los Alamos at least) post WWII into at least the mid 50s. There didn't appear to be any leaks of information regarding hydrogen bomb development or design - the Teller-Ulam configuration from all evidence did not leak to the Soviets (or anyone else, the US didn't share it with anyone) and Andrei Sakharov (re)conceived of it independently a few years later.
The Rosenbergs were handlers who helped get information from Los Alamos (Fuchs, Greenglass, and the possible mystery third leak) to Soviet contacts in the US. What was not revealed at their trial, and not until decades later is that the US had a partial code book (recovered from Finland IIRC) which we used to partially decode Soviet diplomatic cable traffic from the late-WWII period. What was played off as flimsy evidence at trial was in reality a solid case, just that the US was not going to reveal that we had decoded that cable traffic.
The book
Dark Sun by Richard Rhodes has an excellent history of Soviet nuclear weapons development along with probably the most comprehensive non-classified analysis of their related espionage activities.
Quoted:
Did they manage to find all the answers on their own, or did someone get the info for them?
In any technical/scientific pursuit it always helps to have all the answers ahead of time. They did experimentation to confirm most all of the information they got, to make sure they weren't subject to disinformation. The Soviet intelligence agencies as well as scientific leaders were amazed that they got all this critical information so freely. Essentially all the critical discoveries and technical details they got via espionage, they just needed to get the industrial base in place to start manufacturing the materials while the scientists confirmed the information. They started theoretical work on more advanced and efficient designs almost contemporaneously with the US (and before their first test) since they started with essentially the same body of knowledge.
Through most of WWII the Soviet government had better knowledge about the Manhattan project than did the US government. Harry Truman didn't even know the Manhattan project existed until he was briefed after being sworn in as President after the death of FDR. It was agonized over whether, and what, to tell Stalin before using the weapons on Japan which was all moot since Stalin already knew more about the program than Truman did. Truman did however more fully recognize the leap in strategic significance of the weapons than did Stalin.