Hawkins Police Chief Jim Hopper (David Harbour) carries a Smith & Wesson Model 66 as his weapon of choice, notably drawing it in "Chapter Five: The Flea and the Acrobat" (S01E05).
Smith & Wesson Model 10 M&P
Jonathan Byers (Charlie Heaton) takes his father's Smith & Wesson Model 10 when he and Nancy Wheeler (Natalia Dyer) set out to find the monster in Myrkwood in "Chapter Five: The Flea and the Acrobat" (S01E05). She later uses it to fight the Demogorgon in "Chapter Eight: The Upside Down" (S01E08). It also appears to be the weapon used to fake Benny's suicide in "Chapter Two: The Weirdo on Maple Street" (S01E02)
Pistols
Beretta 92FS
The Beretta 92FS appears to be the sidearm carried by the Military Police personnel at Hawkins National Laboratory. While the Beretta 92 is period correct for Season 1's setting in 1983, the use of the Beretta 92FS model is anachronistic in both period availability which wasn't until 1984 and in use with the U.S. Military Police since the M9 Beretta pistol was not adopted by the U.S. military until 1985 (and did not enter service widely until 1990). It would be more accurate for the MP's to be issued with the Colt M1911A1 like the government agents. The majority of these pistols in the series appear to be replicas.
Colt M1911A1
The Colt M1911A1 is carried by government agents including Connie Frazier (Catherine Dyer). Chief Jim Hopper (David Harbour) is issued one when he and Joyce venture into the Upside Down in "Chapter Eight: The Upside Down" (S01E08)
Submachine Guns
Heckler & Koch MP5K
Heckler & Koch MP5Ks with Navy trigger packs, sometimes fitted with a flashlight, are used by government agents and MPs throughout the series. Jim Hopper uses one in "Chapter Eight: The Upside Down" (S01E06).
Heckler & Koch MP5K (Mock MP5K-PDW)
Heckler & Koch MP5Ks with PDW stocks attached are also used by agents. Since the show takes place in 1983, this is considered anachronistic since the MP5K-PDW variant was not developed until 1991. Some of the stocks actually look to be UMP-style stocks, making them even more anachronistic.
Rifles
Mossberg 144 LS
When Will Byers (Noah Schnapp) is running from the Monster in S01E01, he takes shelter in the shed and loads up what appears to be a Mossberg 144 LS.
Shotguns
Over/Under Shotgun
A Double Barreled Over/Under Shotgun appears on display in the "hands" of a mannequin in the hunting store Nancy Wheeler (Natalia Dyer) and Jonathan Byers (Charlie Heaton) go to buy supplies and traps from in "Chapter Five: The Flea and the Acrobat" (S01E05).
As Nancy Wheeler (Natalia Dyer) and Jonathan Byers (Charlie Heaton) go to buy supplies and traps for the Demogorgon, several rifles are seen on a rifle rack in the hunting and camping store in "Chapter Five: The Flea and the Acrobat" (S01E05).