Background/Objectives: Staphylococcus aureus and Staphylococcus pseudintermedius are major opportunistic pathogens in both humans and dogs. In pets, the dissemination of methicillin-resistant isolates (MRSA or MRSP) is problematic for the treatment of animals and is a public health issue due to their zoonotic potential. MRSA and MRSP may also harbor virulent genes that increase their dangerousness. This study aimed to assess the prevalence of (MR)SA and (MR)SP in healthy dogs and their owners in Algeria. Methods: Swabs were collected from various body sites of healthy dogs (n = 88) and from the nose of their owners (n = 38). Antimicrobial susceptibility testing was performed by antibiograms according to the disc diffusion method, and clonality was assessed using Pulsed-Field Gel Electrophoresis (PFGE). All methicillin-resistant isolates were short-read whole-genome sequenced using the Illumina technology. Results: 26 S. aureus and 17 S. pseudintermedius isolates were respectively collected from 13 dogs (13/88, 14.8%). No MRSP isolate was detected, while MRSA was found in six dogs (6.8%). Isolates belonged to ST1 (n = 3), ST 80 (n = 1), and ST 22 (n = 2, including the single-locus variant ST7118). All MRSA displayed the immune evasion cluster (IEC) type E. The ST80 isolate presented the Panton–Valentine toxin, and the ST22/ST7118 isolates carried the tst gene coding for the toxic shock syndrome toxin. Conclusions: The epidemiology of MRSA in healthy Algerian dogs mirrors the one in Algerian people. This poses a zoonotic and public health concern due to the virulence and resistance genes displayed by these isolates. Our results indicate the need for developing One Health strategies to avoid a large-scale dissemination of MRSA in Algerian dogs.