I would not expect that to work, bc the roots would be too bunched up for the trees to grow well or be very healthy. Not to mention how long it would take.

Here's an alternate idea. In the Creasy's Complete Book of Edible Landscaping, there is a pretty lengthy section on espaliering trees. One of the espalier types is called a Belgian Espalier fence. In this design, you plant the trees free standing about 2-3 feet apart instead of against a wall. As the grow you weave the branches to form sort of a harlequin pattern.

Since your goal seems to be privacy maybe you could try something similar with some sort of evergreen instead of a fruit tree. Of course, once it gets over 6-8 feet it would really be a pain to keep training it. If you didn't keep at it, I expect it would turn into a real mess.

Would you settle for a tree every three feet or so and graft the branchs from each tree into its neighbors branches?

It's called a hedge.
Many plants/trees can be used. European beech has been/is used widely in the UK & Europe. I've also seen mention of mixed species hedges - beech, maple, hawthorn, etc.
Just don't plant privet, please.

Debbie, I've seen trees grown this way in Europe. I think it's called "pleaching." The trunks don't actually grow together, but they're planted fairly close together and the branches are woven together as they grow. Would require some pruning and tying.

I've also seen a row of trees planted about 3-4 ft apart, with the branches pruned so that they blend together into a long rectangular shape. Like a raised hedge on trunks, underplanted with seasonal flowers. Very pretty.

sure.. you'll be better trying to graft the limbs... or you might try burrying black locust stumps in the ground together.. like a stockade fence.. these ..even in large diamemeters wll try and sprout... what you are talking about is approach grafting.. you might check out the old artist of trees.. and what is now caled the tee circus in CA.
http://www.arborsmith.com/treecircus.html
I tried an apple fence with limbs grafted ... but sold the place before they were real established.. and the bew owners dug it up in favor of an instant remedy...
Gordon

Actually, it is called pleaching, and it I think it is just beautiful. You need to refer to some pretty detailed books, but I believe you have to cut the trees to allow for the center to meet, and then keep them together until they join. You can create many different designs.