These are 'theories about conspiracies' rather than 'conspiracy theories'. The difference is that the latter is generally derogatory term reserved for the theories that claim to explain a complex idea even if they are based on very sparse or even non-existent evidence that is additionally not backed by a specialist knowledge on the part of the author. The former, on the other hand, are more similar to the 'educated guess' based on factual, albeit usually sparse evidence and are par of the course in the history of espionage or pretty much any context dealing with classified data, where ascertaining validity of most claims is impossible due to inherent secrecy.
That said, the implication that the intelligence agencies of the Warsaw Pact countries were involved in the assassination, although, obviously, not proven due to the aforementioned secrecy that might have led to destruction of evidence, is nevertheless based on various clues and circumstantial evidence, and thus not a product of a vivid imagination of people who read one spy novel too many. Corroboration of these allegation is difficult, as it is assumed that roughly 40% of documents ever produced by SSC have been destroyed in 1990, following to the Order ІV-68 issued by Gen. Atanas Semerjyev, Minister of Internal Affairs.
The possible involvement of the Bulgarian authorities (also known as 'Bulgarian angle') were brought to international attention by the failed assassin, Mehmet Ali Ağca himself. During interrogation, he claimed that the met a Bulgarian national, Todor Ayvazov in 1980 and the latter allegedly gave him the plan of the assassination and getaway. In addition, he named Sergey Bayramik, a Bulgarian national, as one of the conspirators. He also claimed to be approached by Omer Mersan, an agent of the State Security Committee (Bulgarian intelligence agency) who provided money, Bulgarian visa and necessary contacts. In jail, Mehmet Ali Ağca was visited by judge Stefan Markov Pyetkov and investigator Yordan Ormankov who were in fact officers of Bulgarian State Security. Lack of adequate screening on behalf of the Italian intelligence in such a high-profile case led to a substantial scandal. Of course, this cannot be construed as the involvement of Bulgarian intelligence into the assassination attempt itself, as Ağca was a known terrorist and member of the Turkish radical organization and the interference might have been linked to some kind of terrorist activity carried by Ağca or his colleagues within the territory of Bulgaria. Pyetkow himself denied exerting any pressure on Ağca, but did not disclose the character of their conversation and confirmed that he speaks fluent Turkish and has been left with Ağca alone.
According to the investigation of the Institute of National Remembrance in Poland, SSC knew that Ağca arrived in Sofia after the jailbreak following his assassination of Abdi Ipekçi, chief editor of a Turkish newspaper. Agents were most likely informed by Abuzer Ugurlu, a professional criminal who commissioned the assassination of Ipekçi and Teslim Tore, member of Movement for Liberation of Turkish Nation (MLTN), an organization Ağca was familiar with (he was not a member, but one of his close friends was). Both Ugurlu and Tore had ties to Bulgarian state security. It is also important to note that Ağca himself could have met Bulgarian or Soviet agents before, as he took part in the militant training in Syria, conducted by people tied to Palestine Liberation Organization that often employed professional help of the consultants (usually intelligence officers) from USSR and other Warsaw Pact countries. According to Paul Henze and Claire Sterling, the nationalist organizations in Turkey, including 'Gray Wolves' were heavily infiltrated by Soviet and Bulgarian spies. In addition, documents presented by Bulgarian authorities to Polish investigators, Sergey Ivanov Antonov, vice president of the Balkan Air airline branch in Rome, known to Ağca as Sergey Bayramik has been confirmed as a member of the Bulgarian CSS.
In addition, there is also the 'East German angle', related to the disinformation campaign allegedly known internally under the codename 'Papst' (ger. 'Pope'). According to Lt. Col. Günther Bohnsack, chief of 1st Division of the 10th Department of Ministry of State Security (MSS, commonly known as 'Stasi'), Bulgarian and East German intelligence agencies coordinated the efforts to create a disinformation campaign portraying Mehmet Ali Ağca as the solitary assailant tied to the extremist Muslim and Turkish nationalist organization. It is conjectured that MSS conducted this operation because East Germans had good network of agents in the Western media and as Germany was a destination of many Turkish migrants since early 1970s, agents had also a good understanding of political landscape and associated attitudes in contemporary Turkey.
In addition, in March 1995, Italian journal 'Toscana Oggi' published a guideline issued by the Central Committee of CPSU in 1979 that ordered KGB to 'use all means available to prevent a shift in politics that has already been initiated by the new Pope from Poland, also those beyond disinformation and defamation' signed by Mikhail Gorbachov, Mikhail Suslov and other members of the Commitee. This, however, is largely construed as the instruction to focus on the link between the clergy and anti-communist sentiments that gained momentum in Poland, and will ultimately lead to the formation of 'Solidarity' movement in 1980. One of such actions could have been the kidnapping and murder of Jerzy Popiełuszko, Catholic priest and unofficial chaplain of Warsaw branch of 'Solidarity' movement, known for his strong, pro-democratic views. No Soviet nationals were ever included in the investigation and there is no evidence that Soviet authorities took part in the attempted assassination of John Paul II .
Official stance of the Bulgarian authorities, exemplified by the 2006 comment by Gergana Gryntscharova, Deputy Minister of Foreign Affairs, is that the 'Bulgarian angle' was not confirmed, as the there were no proof it the involvement of Bulgarian state agencies and the only Bulgarian national accused of participation in the assassination attempt, Sergey Antonov, has been exonerated by Italian court in 1986.
So, to sum it up, the potential involvement of the Bulgarian and Soviet nationals was mentioned by Mehmet Ali Ağca from the start, although due to lack of evidence and partial withdrawal of the previous statements by Ağca, it has never been properly assessed of confirmed.
Grajewski, A. Skwara, M., Papież musiał zginąć - wyjaśnienia Ali Agcy [Pope had to die - testimony of Ali Agca], Institute for National Remembrance, Katowice 2011.
Henze, P.B., The Plot to Kill the Pope, Macmillan, New York 1985.
Sterling, C., The Time of Assassins, Henry Holt & Co., New York 1983.