What version of SharePoint do you have? If you have SharePoint Enterprise then you will have "InfoPath Form Services" which is a browser version of InfoPath.
If you have the browser version of InfoPath your scenario should work. You will want to add a people picker to your form and then promote that field so that your view in SharePoint will show forms assigned to [ME] so that when a user goes to the site/view they will see the new forms wating for them to fill out.
Ted