If you're going to connect two sections together with a commercial screw connector, for example, then the best way to make sure the connection is clear and aligned is to take one long section, and saw it in two. The angle of the two faces will naturally be aligned at the right angle for each other. A relatively cheap way of ensuring the two sections have their center points aligned involves a pencil and a set or even just the right size of these
http://www.leevalley.com/us/wood/page.aspx?p=44995&cat=1,180,42288&ap=1
Before you make the cut, use the pencil and draw a "V" shape that bisects the cut line. What this does is give you a two-point reference to re-align the two sections. Make the cut using as fine a saw as you can. Japanese saws found in the big-box home centers will make a nice clean cut, but might take a while on green wood or dried hard wood.
The hard part is to find the relative center of the first section. Depending on the connectors and how irregular the circle of the stick you're working with, it may be a matter of "close enough". But once you decide on the center for the one section, then drill the hole. Insert the proper sized dowel center (link above) in the hole, and then align the two sections using your "V" alignment marks. Press gently together and bob's your uncle, you have the aligned center point of the second section.
it's cheap, doesn't require a lathe, and can be used with relative success on a more irregular, "natural" stick, not just a uniformly round one.