Okay, I'm going to answer this for the GDR specifically.

The thing that must be understood is that there WERE actually several anti-Soviet movements (although not always coups) in the Warsaw Pact in the 1950s and 1960s. By all means skip this bit if you know much about any of this stuff; it's just some background.

There was a large uprising in East Germany in 1953, in which hundreds of protesters were shot.

The Hungarian Revolution of 1956 was arguably the most relevant; the revolution succeeded, if only for a matter of days, because significant portions of the Hungarian Army, and even small numbers of garrisoned Soviet troops, defected to the rebels.

At the same time, Władysław Gomułka, a relatively liberal communist, became the first secretary of the Polish communist party. This was deeply unpopular with the Soviets, but the Hungarian issue had already drawn a great deal of international condemnation, and therefore it would have been almost political suicide for the USSR to have intervened there.

This is important because, in 1968, Alexander Dubček became the first secretary of the Czechoslovak Communist Party. His liberal reforms were equally, if not more unpopular, than the Polish ones a decade before. This prompted the Soviets to remove his government from power in the now infamous Prague Spring. By this time the Soviets had more or less figured out how to go about keeping control without as much high-profile activity, and the Prague Spring was much less bloody than the Hungarian Revolution.

Anyways, on to your question.

The events of the '50s prompted many Pact member states to form separate military organizations, with loyalty to the communists, should there ever be another coup attempt.

In East Germany, this took the form of the Wachregiment (special unit) Felix Dzierzynski, which was the military wing of the Stasi.

To become a member of this unit, you had to be both pretty good at whatever your military job was, and also be a loyal communist. Then, if you impressed the people in charge, you might be approached with a job offer. This was especially effective because, not only did it provide the government with loyal military support, but it also ensured that the best and brightest were on their side should things go south.

This wasn't the situation in most places, however. Most countries relied more on paramilitary "militias" to protect key government holdings until external support could be called in. Czechoslovakia had an organization called the Lidové milice (or LM). Unlike in East Germany, where the loyal force was Government-run (East Germany did a lot to make it seem like they had actual democracy), the LM was Party-run. Hungary had the Munkásőrség. The USSR is more difficult to answer for, because it was waaaaaaaaay bigger and had a much more complex structure, but relied primarily on "internal troops (MVD)," which was a mixture of elite special forces, police, and a militia called VOKhR.

Beyond that, coups were prevented by more subtle means. A mixture of positive (Party representatives in the military, perks for communist party members) and negative (surveillance and prison) reinforcement to keep the population loyal. The governments learned how to recognize dissent and quash it early.