I usually answer questions about the Eastern bloc and the economic, social and political situation in the satellite states (especially in Hungary as I'm Hungarian), and this is a very good question.

I have to mention here that it's very important that you wrote 1961. Every decade from 1947 to 1989 was VERY different. 1961 is a good start. We are 5 years after the crushed revolution, Kádár had already consolidated his power. This is very important, because he was pretty much hated in 1956 (he was considered as a puppet of the Soviets), and by now he had achieved a sort of social peace.

It was just around 1961 when the leaders of the Communist/Socialist party (called MSZMP) realized that the current economic policies were simply bad and reforms were urgently needed. They developed a new policy called New Economic Mechanism (NEM). It was made by reformer Communists and economists, they showed it to Kádár and he gave it a green light after he made sure that it was compatible with the Marxist-Leninist ideas and it was acceptable by Moscow. Brezhev allowed it, because in the Kremlin the reformers were more influential and they could convince him to leave Kádár alone.

After it was implemented in 1968, the NEM proved to be quite successful, the long oppressed private sector got some space, the large state factories somehow got closer to the realities of the market (and further from the rigid planned economy).

However in 1972 it had to be stopped.

The reason? First, the NEM started to grow too big, hardliner Communists thought it would lead to democracy and Capitalism. Second? After the Prague Spring in 1968, hardliner Communist sized the power in Czechoslovakia, in 1970 the same happened in Poland, and in the Soviet Union, there was a so called re-Stalinization lead by Brezhnev. By that time the anti-reformists became the stronger influence group in the Kremlin, and they considered the NEM too Capitalist and anti-Marxist, so the Kremlin (practically Brezhnev) told Kádár in February, 1972 to stop NEM immediately. (Hardliner Communists almost managed to pull off a coup d'eta against Kádár in the same months because they opposed NEM so much and complained to the Kremlin - by then lead by anti-reformists).

So basically what happened in Moscow, it was mirrored by the satellite states. There is a joke from the era, which makes sense if you know that we use the same word for "steering wheel" and "government" in Hungarian.

What's the longest car in the world? Hungary. The car is in the country, but the steering wheel/government is in Moscow.

EDIT: Answering your example, relaxing press censorship:

If in the MSZMP and in Moscow the reformers were stronger (we are still talking about 1961, because for example in 1981 it would have been different), then Kádár could easily relax press censorship. Of course, the more influential part in this question is Moscow.