Yes, the spike protein in vaccinated persons if secreted into extracellular space may be presented in secretion such as sputum or saliva.
“the protein may also be secreted into the extracellular space, where it may be recognized by B cells (which make antibodies) or taken up by antigen presenting cells and re-processed. The protein may be found on the surface of the cell in either its peptide form or its native form, likely until the cell dies or interacts with other immune cells.” [1]
Spike proteins created by host cells can bind to ACE2 receptors on our cell tissues and caused immunopathology to the tissues just like the protein from coronavirus.
“it is indeed the binding of the Spike protein to ACE2 that's a key part of this process.” [2]
If the spike proteins bind to heart muscle cells they may cause myocarditis or pericarditis.
Study: Immunization with SARS Coronavirus Vaccines Leads to Pulmonary Immunopathology [4]
Suppression of information regarding to COVID-19
https://www.indiatoday.in/technology/news/story/mrna-technology-pioneer-says-covid-19-vaccinated-people-can-shed-spike-protein-twitter-says-delete-this-1809062-2021-05-31
Footnotes
https://www.idsociety.org/covid-19-real-time-learning-network/vaccines/vaccines-information--faq/
https://www.science.org/content/blog-post/spike-protein-behavior
https://www.researchgate.net/post/Can_the_spike_proteins_excreted_by_vaccinated_people_cause_serious_long-term_damage_to_nature_and_the_environment
https://emergence.quora.com/Study-Immunization-with-SARS-Coronavirus-Vaccines-Leads-to-Pulmonary-Immunopathology-Lesson-Learned-from-Challenge-wit
Peter McCullough MD interview post on Twitter Jan 26, 2023
"Shedding Concerns: "It Looks Like the mRNA is Transferring from the Vaccinated to the Unvaccinated"
Could you actually take a vaccine inadvertently through close contact, kissing, sexual contact, or breastfeeding? Peter McCullough MD says, "it looks like the answer is YES." "
https://twitter.com/HappyCamper2626/status/1618803494076710913