This answer is an expansion on ohwilleke's answer, which explains that the reason probably involves avoiding conflicts of interest with other clients.
The relevant provisions in england-and-wales are to be found in the Solicitor Regulation Authority's Code of Conduct for Solicitors, RELs, and RFLs which has statutory force by way of sections 31 and 32 of the Solicitors Act 1974, section 89 of, and paragraphs 2 and 3 of Schedule 14 to, the Courts and Legal Services Act 1990 and section 57(2) and (8) of the Legal Aid, Sentencing and Punishment of Offenders Act 2012.
Conflict of interest
Rule 6.2 of the Code of Conduct states:
You do not act in relation to a matter or particular aspect of it if
you have a conflict of interest or a significant risk of such a
conflict in relation to that matter or aspect of it, unless [... exemptions omitted ...]
"Conflict of interest" is defined as follows:
[...] a situation where your separate duties to act in the best interests of
two or more clients in relation to the same or a related matters
conflict.
Own interest conflict
Less likely, but still possible (and therefore still a necessary component of compliance checks) is an own interest conflict (e.g. the solicitor's personal affairs are linked to the landlord in some way). In that case the relevant rule is 6.1:
You do not act if there is an own interest conflict or a significant
risk of such a conflict.
An "own interest conflict" is defined as follows:
[...] any situation where your duty to act in the best interests of
any client in relation to a matter conflicts, or there is a
significant risk that it may conflict, with your own interests in
relation to that or a related matter.
"Where it gets sent"
Ordinarily, it should not get sent anywhere without your permission, pursuant to rule 6.3:
You keep the affairs of current and former clients confidential unless
disclosure is required or permitted by law or the client consents.
This is the case even if the solicitor doesn't accept you as a client. That's because "client" is defined as (emphasis mine):
the person for whom you act and, where the context permits, includes
prospective and former clients.
There is also the common law principle of legal professional privilege which may protect disclosure by a solicitor of confidential information that a non-solicitor might otherwise have to disclose by law. However, to be applicable the information must have been communicated in relation to legal advice or litigation. The rules are a little complex and you can't assume that everything you tell a solicitor is protected by privilege.
Privilege and confidentiality under the Code are separate and overlapping rules, so what isn't protected by one might still be protected by the other. The main differences between the two are that (a) the law can override confidentiality but not privilege, (b) confidentiality is a duty (not to disclose) while privilege is a right (to avoid disclosure). The first difference is because of the exception for confidentiality (in rule 6.3 above) where "disclosure is required or permitted by law". Contrast that with e.g. Bowman v Fels [2005] EWCA Civ 226 at paragraph 90 where the Court of Appeal held that the Proceeds of Crime Act 2002 was not capable of overriding privilege.
Examples of disclosure being required by law are: