What would happen if two people shared blood when they're not the same blood type?

yes, the antibodies created by the blood will attack the hosts blood and the hosts blood will attack the foreign blood cells

My biology teacher always told us that if you did this then the person recieving blood would probably die from agglutination. This basically means that red blood cells would clump together and is a result of the antibodies interacting with antigens.

The Bloodtypes are defined by the antibodies and antigenes present. They are grouped after a system by Landsteiner (the ABO system)
Group A has antigenes A and Antibodies for antigene B
Group B has antigenes B and Antibodies for antigene A
Group AB has both  antigene but no antibodies
Group O has no antigenes but both antibodies

as antibodies will react with their respective antigenes and cause the blood to coagulate you can only combine full blood with the same bloodgroup.  In medicine antibodies are removed so you can combine:

donor:
A  gives A and O
B gives B and O
AB gives AB and O
O gives all