The electronegativity does change across the periodic chart as well as up and down. However, if you move along the chart correctly, you can find atoms that have the same electronegativities.

Iodine and sulfur, for example, have the same electronegativity. If you start at Iodine and move up, you are increasing electronegativity. However, as you move left to get the sulfur, you are decreasing electronegativity. The two effects exactly cancel out in this case. In other words, the amount that you increase as you go up the chart is exactly the same as the amount you decrease by moving to the left on the chart. As a result, both atoms have the same electronegativity.

In the compound I₂S, then, each I shares electrons with S. Thus, they each have a covalent bond with S. Since the electronegativities are the same between the two atoms, however, each pulls on the electrons with the same strength. As a result, the electrons are shared perfectly equally, forming a purely covalent bond.