I don't know about the others, but I am familiar with the Perry expedition.

While the Japanese did forbid anyone entering or leaving the country, there were exceptions and there were accidents.

The main exception was the Dutch in Nagasaki. Throughout the Edo period, the Japanese made a special dispensation for the Dutch to land and engage in a limited amount of trade at the small island of Dejima. They got this special exemption mostly by virtue of not trying to convert the Japanese to Christianity like the Spanish and Portuguese. This meant that there was a small contingent of Dutch-Japanese translators maintained by the Japanese Shogunate.

There was also a man named Ranald MacDonald who deliberately shipwrecked himself on Hokkaido to see Japan. The Japanese sent him to Dejima, and while there he took on 14 students and taught them English.

There was also a number of shipwrecked Japanese sailors who had wound up in the US. They taught a few people some Japanese, and one of the ones who made it back to Japan would go on to become a top governmental adviser.

Both nations also had relations with China, which meant that Chinese, too, could be used as a lingua franca.

So when the Perry expedition was launched to open negotiations with Japan, they brought Samuel Wells Williams, an American linguist who lived in China and had spent enough time with a group of shipwrecked Japanese sailors to pick up a basic level of Japanese, and Dutch-born American diplomat Anton L. C. Portman. The Japanese, for their part, managed to scrounge together their own group of interpreters.

The final treaty was transcribed and signed in English, Dutch, Chinese, and Japanese, with both sides having say in the wording of each translation.

While this is just one case, it does show how it's quite common for two cultures to be able to find a common language with a little effort. The only time I can think of off the top of my head where there would have been no way to break the language barrier from the start would have been Columbus's first expedition. Once they could communicate with a few Native Americans after that they could use them as interpreters with the other tribes. By the time the Pilgrims founded a colony at Plymouth Rock, the first Native they had contact with already spoke English.

As for when there isn't an interpreter or lingua franca handy, it starts with a mix of body language and mutual learning of each other's languages, resulting in the creation of a pidgin language that combines the elements of the two.