Yes, in order to apply for a Work Permit for you, an employer needs to show that there is no suitable local person (i.e. someone settled in the EEA or Switzerland). But if the job is in a shortage occupation, there is no need for them to have to show this.
You do not need to return to the US to switch from student status to Work Permit holder, you can do it in the UK (if you find an employer who will employ you before your student visa runs out, of course).
What's more, you may not need a Work Permit at all. If you complete a postgraduate course, you can now stay and work in the UK for 12 months without a Work Permit under the SEGS scheme. When the 12 months is up, you need to either leave the UK or switch to a status that allows you to continue to work, eg. HSMP or Work Permit holder.
This is an overview. There is more about SEGS, Work Permits and HSMP at
www.workingintheuk.gov.uk