It is possible to solder both sides when IC sockets are used. I use the machined-pin type, and this board was completely soldered on the top. Those pins that stick through could have been soldered on the back layer too.
Edit: If you use the machined-pin carriers without the plastic, it is even easier to solder both sides and still have a socketed IC.
You just need a real thin tip. I have also done other top side soldering by heating the wire/pin from the bottom and feeding the solder from the top. Pre-tinning of the pad helps, but is not required, particularly when the wire is heavy. I used that technique for some larger bridge rectifiers on an otherwise SMD board.