SBM
I know from personal experience that in my state, emergency ROs are issued on the basis of allegations alone. No supporting documentation needed.
The judge examines the petition to make sure that the allegations fall within the parameters set out under law for emergency ROs; if they do, the order is automatically granted.
If the respondent chooses to challenge the RO, then the petitioner has to present evidence to the court (a show-cause hearing).
If the respondent does not choose to challenge the RO, then the emergency restraining order automatically becomes a lasting restraining order.
I know that many other states operate the same way.
My example would be the 2005 David Letterman case, where
a woman filed for and received a temporary RO against David Letterman, prohibiting him from coming within three yards of her.
She alleged that he was using code words, gestures and eye expressions during his TV show to convey secret messages to her.
http://www.people.com/people/article/0,,1144343,00.html
I doubt that the judge in the case really believed the petitioner but the judge's beliefs are not called into question at the time an emergency RO is issued.
I know that the reason my state (which is not New Mexico) does it this way is to balance the needs of victims against the rights of respondents. A victim of domestic violence may need the immediate protection of a restraining order but not have enough time to immediately gather the supporting documentation.
The respondent's right to freedom of movement and association should not be unduly abridged, so the respondent is given the guarantee of a swift hearing if they choose to challenge (in my state, the show-cause hearing must happen within ten days after the respondent files to challenge).
It's not perfect solution but it's the best that legislators could come up with.