Hello.

~ sign means relative URL and usually it is replaced by site´t root (domain), so you get full URL. So the problem may be related to how your localhost site is configured, or to the fact you probably connected database to your web site somehow manually (maybe you just changed connectionString, I don´t know).

If you have second instance where you site is running fine and you want to move it on your local machine, I would recommend you to use export / import functionality. You can simply export your live site and import it on your local machine (into some existing and running instance). Everything should be fine then. Could you please try it?

Best Regards,
Radek Macalik