Any details on P. Westergaard furniture?  

Based on the title,  P. Westergaard was the furniture factory (literal meaning of Møbelfabrik).  The doesn't really rule out any of the other roles you mentions, but would sort of suggest against them.
The drawers look very nicely executed.  Case looks well executed as well.

As Leif says "P. Westergaard . Møbelfabrik" is the manufacturer.
Here's a little additional info.
"Silkeborg" is the city in Denmark.
"Telefon 3888 fl. lin." = "phone 3888 multiple lines."
The print in red "nr 14 palis" must mean "model no. 14 rosewood" (palis=short for palisander=rosewood in danish)
The black print "bil" could either be "car" (delivery method), or short for "voucher".
"Fa." short for "Fabrikant"="Manufacturer". I wouldn't put to much credit into that title. In Denmark it was more well esteemed to be a "Manufacturer" than just a "Furniture retailer".
"Frøkjær Knudsen" the name of the retailer.
"Møbelmagasin"="Furniture store"
"Asylgade 9  Thisted" street and city of the retailer.
And here's some additional additional info.
The furniture is crafted before 1967. Denmark got postal code/zip codes that year. There is no code on the label.
I'm not an expert in the old phone numbers, but I believe the 4-digit numbers were used from the 1940's-early 1960's. (today we got 8-digits in Denmark.)
And to round it of (if you're still reading), here's some additional additional additional info.
The retailer, Frøkjær Knudsen, continued to work until the age of 90. Hence the Furniture store existed until the year 2000. On the picture below you can see his shop, under the sign saying "MØBLER".
Despite the, more or less interesting, historical facts, I have absolutely no idea who actually designed the nightstands.
 
 
 

Actually that analysis of the label is fascinating. Thank you for it.  
And, unfortunately that is probably all that will be known until somebody turns up some vintage documentation on P. Westergaard.  
It is entirely possible that if any documentation is to turn up, it would prove to be a generic piece.  And if it were not to be strictly generic, it is quite likely that the designer would be an unrecognized name, or barely recognized.  
Having looked through vintage Danish department store / mail order catalogs, I was struck by just how many pieces were generic.  There were occasional Wegners (esp. the Wishbone chair) and Mogensens that were credited or there were uncredited Møllers and Kai Kristiansens and Folke Ohlsens et al, but the vast majority was generic.  

Thank you I_Love_danish and Leif, definitely more info then I expected, and some very useful tips. 
I will keep an eye out but am happy not knowing, they are in very good condition for pre 1967, possibly refinished.