I’m not sure I understand what you mean by “sticky”, do you mean the button stays in the same position in the browser window regardless of where the user scrolls on the page? This is done with CSS, something like this in style.css:
.sharing {
position: fixed;
left: 0;
}
The left dimension may need to be tweaked slightly and you may need a top dimension, the key is the fixed position.
I’m also not sure where in your site you want this. All pages? Home page? Search page? Archive page? Each page has a separate template file on which to place the code, but if you want it on all pages you can place the code on header.php. Where on the file might be determined by where you want it to initially appear, but if you specify a top dimension in CSS it will not matter much as long as you do not cause a syntax error by placing the HTML in a PHP block or something. Try at the very bottom of header.php below <div id="main" class="wrapper">
If what I’ve suggested does not quite do it for you, don’t be afraid to try different locations or CSS. Just be sure to have a copy of the original file in case something you try goes badly. As long as you do not put HTML in a PHP block, it cannot go that badly.