We have a few examples of people carrying and using two weapons at once in the sagas.

In Njáls saga, Gunnar Hámundarson fought several times with an atgeirr in one hand and his sword in the other. In Egils saga, Egill and his brother go to the battle of Brunanburh with a kesja and two swords each - one attached to their wrists via a cord, and the other on their hip. Presumably, these swords on their wrists were fairly small.

The way these were used, however, wasn't quite what you expect from video games or RPGs when it comes to dual-wielding, however; in Njála, Gunnar used one to parry and the other to attack, but never attacked with both at the same time, while in Egla, the Skalla-Grímsynir used their additional weapons in a cycling style; using their kesjur at reach, and swapping to swords in closer combat, while carrying their shields.

Speaking of shields, there's an argument to be made that they were used as weapons in early medieval Scandinavia, just based on the biomechanical usage of them, as they lack stabilizing straps for the forearm.

You may want to cross post your question to /r/wma as this is a pretty hot topic with people who participate in HEMA(Historical European Martial Arts).

In short, yes people did practice dual wielding, although the extent of the wielding may have been relegated to the classroom and/or exhibitions.

This is a good video that talks about it: http://youtu.be/jZNZyhNFSaE. The gentleman speaking is Matt Easton, who is an expert in historical European arms and armour.

He mentions Manciolino, who's entry at http://wiktenauer.com/wiki/Antonio_Manciolino may provide some more info about the man himself. Matt mentions Asia at the end which has more examples than Europe.

There are certainly even more examples than Matt mentions in Asian martial arts. Eskrima for example is two sticks. You can find a bunch of videos on youtube in regards to that, even some full contact videos which can get pretty crazy.

Sorry to any mods, I know this is not up to current answer "snuff", but I thought mentioning HEMA and the wiktenauer website were important to the subject.

edit:

I would add that many that study HEMA and original training texts have an understanding that a shield is not only a defensive tool, but a weapon as well. Much like /u/EyeStache said. You can actually watch a number of HEMA related videos on youtube and see some shield edge strikes in live competition.