DBS check certificates, of all levels, are sent to the subject of the check and are to be retained by them. (A notification of the result can also be sent to the requestor.)
Indeed, so the applicant / student gets a big green piece of paper, we are told that the paperwork has been sent by DBS or Disclosure Scotland (it is more complex in NI and the Crown Dependencies) and then we expect someone to present it physically for checking. The paper has various design features on it to prevent forgery. We get to hear that DBS has closed the process, so I'm waiting to get the paperwork. I don't actually get direct advice from DBS normally, but very occasionally I would get a call or email from someone in DBS or a police force asking me to ensure the matter is fully checked, though more usually to clarify the job role scope. Obviously if the paperwork isn't presented within a few days, or I get a phone call, I tend to get a bit twitchy.
Before that has happened, well before, and this is a point I so love to make at this point, the applicant or student will be typically asked in writing as well as verbally something along the lines of "is there anything we, as employers / educators / sponsors need to know?" or "kindly disclose all cautions and convictions", with some verbiage about convictions received when under 18 years old. And lying about that is far, far more worrying to me than saying "I dodged a few rail fares a few years ago" What else are you hiding?