Is every pad in eagle a via?
In normal two-sided PCB production, all holes, whether vias or holes for component pins, will be plated to connect both sides of the board, unless specifically declared non-plated, and a separate drill file provided to the board maker..

Of course, if you are making the board yourself using normal home etching techniques, no holes will be plated, and you will have to solder wires in holes where you want connections between the top and bottom layers.

If the board shown in your question is home-made, then you will have to solder both sides to ensure proper connections.  If the board is commercially-made, the holes should be plated, so you will only need to solder the pins on the bottom side.