Well I just encountered the same challenge with a ton of Stickies on my G5 OSX Tiger 10.4.11 that I wanted to move to my new iMac 27" running OSX Mt. Lion.
Open a Mt. Lion Finder window and select your Home user and then holding down the Option key select GO in the Finder menu and select LIBRARY. (Library won't show up in the list without the Option key held down.) Look down the list for "StickiesDatabase" file. Highlight it and from the Finder menu > File > Select DUPLICATE to create a backup copy of the empty database just in case.
If your machines are networked together open a second Finder window and connect to your old machine. Follow the same path Home user > Library > and find StickiesDatabase file. Click and drag it into the other Finder window and drop it on top of your new empty StickiesDatabase file name. A Requestor comes up saying a newer version exists, do you want to replace......select replace. Bingo, all your stickies are now transferred and even maintain the formatting on the text.
If your computers aren't networked just copy the old StickiesDatabase to a JumpDisk or CD disk and move it to the new machine that way. Drag the file on top of your Home user > Library > StickiesDatabase same as above.