Reply from Arun:
If you are using a custom component, then I think you can have different ids. and based on search input, you can trigger an Ajax request to get autocomplete results with q query parameter in both the cases. It will be just input filed id which is different.
<form name="searchform" method="get" action="results.html" target="_blank">
<input type="text" id="header--serach-input" name="q" />
<input type="submit" value="Search" />
<div id="autocomplete"></div>
</form>
<form name="searchform" method="get" action="results.html" target="_blank">
<input type="text" id="footer--serach-input" name="q" />
<input type="submit" value="Search" />
<div id="autocomplete"></div>
</form>