A recession in the United States could certainly trigger a recession abroad, just as economic problems in other countries could cause problems in the US.
The world economy is extremely globalized today. Other countries are tied to the US economy because they depend on exporting to the US. If demand in the US drops because of a recession, jobs connected to exporting can be lost. There are also financial connections with people in other countries having investments in the US. Because of all these types of connections, recessions in one country, like the US, can trigger recessions in other countries.