updated answer: The .380 Remington Arms UMC [Union Metallic Cartridge Co] pistol is the semi-automatic centerfire model 51 pocket pistol, also made in .32 in much smaller numbers, with production of the two calibers totaling about 65,000 from 1918 - 1927. The model 51 was designed by famed weapons designer John Pedersen, and all model 51 s/n start with "PA" for "Pedersen Automatic". They are highly regarded for their ease of use (although not ease of disassembling and reassembling).
Primary US competitors were the Savage Arms pocket pistols (models 1907, 1915 and 1917), made 1908-1928, also in .32 and .380, and the Colt pocket pistols.
original answer: Pistol? Semi-auto Model 51. No pistols.
Manufactured 1919
Condition?
Depends on condition, condition, and condition. From $100 to $1,200 LNIB.
Greatly dependent on condition. from $200 to $1,200 for new in the box.
Depends on original condition from a $100 parts gun to as much as a $1,200 new in the box.
The value will be determined by the condition of the pistol. Anywhere from $0 to $500. In Mint condition especially in the box they can be worth double that.
The value of a Remington UMC 32 caliber pistol actually depends on a number of factors. Some of these factors include the age and condition of the pistol.
The Remington Model 51 pistol was not made by Walther. Has nothing to do with Walther. See the link at the right for a reproduction manual. sales@countrygunsmith.net
According to the Remington website, there are three possible years for the manufacture of your Remington UMC 380, with a serial number of PA 50795. Those years are 1995, 1968, or 1923.
http://stevespages.com/ipb-remington-51.html
What Model 51 pistol or a rifle and what model?
Someone at the Remington Society of America web site could probably help you.