Personally, I'd find it very irritating to have neighbours either side "chatting" regularly across my space, but for the sake of neighbourly relations a high fence might be a step too far.
Our neighbours on one side had a four foot fence with about a foot of trellising on top - not too exposed for either of us but we could see if someone was in the garden if we wanted to communicate, but weren't on top of each other. When they sold their house, the new neighbours immediately put up a 6 foot solid fence. Our garden is about 9 inches below theirs (we live on a hill) so the effect was like the Great Wall of China and put a large slice of our garden in complete shade after about one o'clock in summer and most of the day in winter! Complete redesign of border on that side of the garden ensued 
and it did feel rather an odd thing for them to do, although we never said a word.
In your situation, I think I would find some shrubs that will grow high enough and quickly enough to create a semi-permeable barrier, enough to screen but not a solid "wall". How about some Philadelphus? Beautiful scented flowers in the summer, leaves in spring until late Autumn. If the budget would allow it, you could get a more mature specimen that would give you the privacy you need more quickly. Or, as other have said, some trellising and grow a honeysuckle, clematis (there are some everygreen ones that grow quickly) or climbing rose up it. You would have a perfect excuse in saying that you wanted flowers and/or scent near the house and you would only have to do it on the one side to have the desired effect of cutting off their direct eyeline in order to chat.
Hope you find the responses here helpful, and that you come back and let us all know what you decide to do.
