I am also curious about the reactions of left-leaning intellectuals (aka "fellow-travellers").
It is difficult to characterize the reactions of whole organizations, and to separate out the reaction to the speech from the reaction to the 20th Party Congress as a whole, and to the other events of 1956.
Many groups were internally split between those who saw it as a call for renewal and others as a betrayal. Because the Russian Revolution had progressed further than any other at that point, many Western leftists looked to the Soviet Union with genuine admiration as a more advanced society. To criticize Stalin and the Soviet Union was to undermine the cause. Others, however, welcomed the opportunity to cast off association with Stalinism. In some cases, the anti-Revisionists prevailed and drove out the reformist elements; in other cases, the reverse happened.
Moreover, 1956 was an eventful year: mass demonstrations against de-Stalinization in Georgia, the Suez Crisis, the Poznan strikes and their harsh response, and most significantly the Hungarian Uprising, which some say were catalyzed by the speech. The violent repressions led many to question Soviet leadership. Indeed, 1956 is a conventional point for marking the rise of the “New Left” (a term popularized later), which favors a more humanistic socialism, and of the communist movement independent of the Soviet Union later labeled as "euro-communism."
For a roundup of some of the major parties:
French Communist Party (PCF)
Maurice Thorez, longtime leader of the PCF, was such an ardent Stalinist that he had broadcast a plea for French troops not to resist the Nazi invasion (as Germany and the Soviet Union were allied at the time). While he was ailing physically in the 1950s, his close associate Jacques Duclos, who shared his pro-Soviet sympathies, was at the helm. As such, reformists such as Laurent Casanova, Maurice Kriegel-Valrimont, and Marcel Servin were purged from the leadership. It was under Waldeck Rochet, after 1964, that the party became more flexible.
Italian Communist Party (PCI)
Palmiro Togliatti, head of the PCI, had virtually since the death of Stalin sought a more independent direction; after the release of Khruschev's speech he openly criticized Stalinism in an interview with Nuovo Argomenti. The PCI did see splits, however, over the Hungarian Uprising.
Communist Party of Spain (PCE)
The PCE was strongly Stalinist and sought to suppress news about the speech, but Spain already had a large anti-Stalinist movement, the Trotskyist POUM, which published it in its newspaper. The situation in Spain was further complicated as many leftist leaders were forced into exile (or worse) under the Franco dictatorship. So, the speech did little to change "facts on the ground."
Communist Party of Great Britain (CPGB)
Party members wrote in numbers following the publication of the public speech, and the old guard paid it little heed; J.R. Campbell declared the discussion on Stalin closed on March 12, only to be undermined when the text of the secret speech reached the West on March 16. Similarly, General Secretary Harry Pollitt expressed full confidence in the Soviet Union and the CPSU at the 20th Party Congress, but by May the defense of Stalin was untenable, and he resigned. Dissatisfaction with Soviet actions and to the quashing of debate on the same led to an exodus of members, including some its leading intellectuals: John Saville writes that he and E.P. Thompson founded The Reasoner (and The New Reasoner after they left the party) in response to the speeches.
Communist Party of Germany (KPD)
The KPD at first hid the speech from its members, but they learned about it anyway from the mainstream press in June. Before much internal turmoil could roil the party, it was banned in August by the Constitutional Court for its more direct revolutionary activities. There was no major communist party again in West Germany for over a decade.
Danish Communist Party (DKP)
The DKP's popularity had waned as the Cold War intensified; seeking to reverse this trend, Aksel Larsen used the speech and the aftermath of the Hungarian repression to reconsider his past staunch support of Stalin. This was ill-received and he was forced out of the party.
The Norwegian Communist Party likewise remained in Moscow's orbit.
Communist Party of the Netherlands (CPN)
I could not locate materials specifically addressing the CPN, which was caught up in a fight over the merger of trade union and with dissatifaction with Paul de Groot.
I wish to supplement choster's answer.
Communist Party of Australia (CPA)
CPA leadership discovered the Secret Speech as the CPNZ leadership transhipped through Australia early in the year. According to the then informal organisation of International fraternal parties, the CPA was dependent on the CPGB for material at this level. They didn't find out until CPNZ leadership had a face to face with CPA leadership. The CPA leadership were horrified by the Secret Speech ontop of 20th Congress. The CPA leadership was highly Stalinised, and had to keep a lid on both Thompson-style reformers, and a left-revolutionary union-based CP vibe. They decided to hide the speech, and the 20th Congress was not heavily discussed in public. When the speech leaked in June through the NYT (IIRC) they policed members vigorously at the Branch level, meaning a leadership directive was issued.
This caused a pre-split situation, when a significant body of what would later be called "socialist humanists" and the like started reacting to their persecution within the Party. The CPA had lost most of its left nationalist fellow-travellers from inside the party over 1945 (Revolution Now), 1949 (Too much Revolution) and various other things. The stayers were there because they had strategic or family reasons to be in party.
Then Hungary 1956 blew the lid on the Australian party's fragment. Helen Palmer founded Outlook, a socialist humanist journal who spent much of its time reconsidering Leninism through Stalinism and through Indigenous liberation issues. The 1956 splitters were heavily reviled, and as late as 2006 were viciously attacked in public by CPA and ex-CPA members. (Somehow this was a "worse" betrayal than leaving the party over 1945 or 1949 or 1968). The party's engagement with Maoism, and with 1968 issues probably relates to the unresolved questions over reformism and the path to socialism that 1956 brought forward. However, I'd suggest that the Secret Speech brought forward the issue of party independence and democracy, and Hungary brought forward the issues of Stalinism.
The Party leadership started looking towards Maoism after this, charting what would eventually become a kind of independent left-reformist-Stalinism conditioned more by the reformist practice of the CPA and its relative union strength as the best militant union leadership body in heavy industry and transport.