The most dominant style guide is The Chicago Manual of Style, which recommends against spaces around em dashes, but there is an AP style guide which actually does recommend spaces!
Em-dashes are not the same thing as parenthesis even if they are used in similar circumstances, and as a sibling comment recognized well, you'd still put spaces outside the parenthesis —but this would be terrible with em-dashes— wouldn't you agree?
This is really a style issue, and it mostly depends on what "school" you subscribe to.