http://www.expatarrivals.com/romania/work-permits-for-romania
mentions that in order to apply for my work permit, the employers need to prove that they have made every effort to find a Romanian national to fill the position that they are now offering to an expat.
How exactly does an employer go about proving that?
It is quite common for companies to have to prove in some way to national authorities that a local cannot do a job that a foreigner can before the company hires the foreigner. Many countries in the world have laws to this effect.
How strict the government is and how easy it is to fake or exaggerate the paperwork will obviousbly vary from country to country. For starters, I imagine the Romanian company will have to show they have advertised the position for long enough without finding suitable candidates within Romania, and that the foreigner in question has some skill that will be good for the position but which no other (Romanian) applicant possessed. They will submit this evidence with the application for a work permit/visa on behalf of the foreigner.
It is often simply a matter of routine paperwork, as I said - it depends on the country.