Bottom line:  Stethoscopes, mostly the earpieces, are a reservoir for bacteria, but there is no evidence directly addressing any association between use of stethoscopes on multiple patients and transmission of the bacteria found on the stethoscope

Halcomb E, et al. Role of MRSA reservoirs in the acute care setting. JBI Library of Systematic Reviews. 2008; 6(16): 633-685.
Identified one observational study of the eartips of stethoscopes dedicated to patients on contact precaution for MRSA.  13 of 78 (17%) eartips examined had potentially pathogenic bacteria on them, but none of the bacteria were the same as the infectious agents in the patients.

Other similar studies were identifed in a PubMed search.   Only the Brook (1997) paper below describes an actual infection (in a nurse) traced to an earpiece of a stethoscope.  The other studies show that most common organisms identified on stethoscopes (mostly earpieces) is S. aureas.

Gopinath KG, et al. Pagers and stethoscopes as vehicles of potential nosocomial pathogens in a tertiary care hospital in a developing country.  Trop Doct. 2011 Jan;41(1):43-5.  PMID: 21109607

Youngster I, et al. The stethoscope as a vector of infectious diseases in the paediatric division.  Acta Paediatr. 2008 Sep;97(9):1253-5.   PMID:18554272

Bernard L, et al.  Bacterial contamination of hospital physicians’ stethoscopes.  Infect Control Hosp Epidemiol. 1999 Sep;20(9):626-8.  PMID:10501265

Brook I.  Bacterial flora of stethoscopes’ earpieces and otitis externa.  Ann Otol Rhinol Laryngol. 1997 Sep;106(9):751-2.  PMID:9302906

Smith MA, et al.  Contaminated stethoscopes revisited.  Arch Intern Med. 1996 Jan 8;156(1):82-4.  PMID:8526701

Wright IM, et al.  Stethoscope contamination in the neonatal intensive care unit.  J Hosp Infect. 1995 Jan;29(1):65-8.  PMID:7738341

Search strategy:
Consulted and evidence summary source:  Searched Joanna Briggs Institute for “stethoscope and infection” – Identified a systematic review, which cited a paper on stethoscopes dedicated to MRSA-infected patients as as reservoirs.
Looked up that reference in PubMed and identified terms for a focused PubMed search:
(“Stethoscopes/microbiology”[MAJR] OR “Equipment contamination”[MAJR]) AND (nosocomial OR “hospital-acquired infection”)

Used QUOSA to identify articles from this set that discuss stethoscopes.