I'm not a chemist, but I asked my officemate who is a chemist, and he says that you're actually asking two questions. The first question is whether the polarity will change if an atom bonds with the same atom from another molecule, and the answer is yes, the polarity will change.
The second question is how can the bond form if the polarity is different, and the answer to that is that electrons are shared or "stolen" when chemical bonds form. Don't confuse polarity with chemical bonds (some molecules have polar bonds but are not polar molecules). If you are only considering a bond between the same two atoms, it is a covalent bond. In covalent bonds, atoms share electrons equally. So when two hydrogen atoms bond together, both share an electron to fill a valence shell.