If a plaintiff and defendant have a personal connection, and one (or both) of them represents themselves, won't that pose a conflict of interest between the councillor(s) and the opposing side(s)?
The purpose of the conflict of interest rules is to ensure that attorneys are acting in the interests of the clients they represent in that particular case and not their own interests, the interests of their friends or families, or the interests of their other clients. If a person is representing themselves, it is impossible for them to have a conflict of interest that would prevent them from serving their own interest because any such conflict could only change their interest.
Say Alice wants to represent herself in a lawsuit against Bob and Bob hires Charlie to represent him. Even if Charlie is a close friend of Alice, that couldn't cause Alice to act against her own interests out of friendship to Charlie. Necessarily, she would only act in Charlie's interests if it was in her own interest to do so.
So there simply cannot be a conflict that would prevent a person from acting in their own interest. They could only put someone else's interest above their own if it was in their own interest to do so, in which case it would not be putting anything above their own interest.
I suspect the OP has misunderstood the concept of Conflict of Interest.
It refers to solicitors - not Litigants in Person - who are acting for (a) a client where there is a conflict of mutual interest or (b) where there are multiple clients who are at odds with each other.
Two very simplistic examples of each, where Aziz wishes to sue his Home Owners Association:
(a) If a solicitor, Zara, is a member of the HOA board then she should not represent him (see para 6.1, below).
(b) If the HOA engage their own solicitor, Bhupal (as Zara is too busy) then Bhupal cannot also represent Aziz (see para 6.2, below with caveats).
The Solicitors Regulatory Authority's Code of Conduct says that:
6.1 You do not act if there is an own interest conflict1 or a significant risk of such a conflict.
6.2 You do not act in relation to a matter or a particular aspect of it if you have a conflict of interest2 or a significant risk of such a conflict in relation to that matter or aspect of it, unless:
(a) the clients3 have a substantially common interest4 in relation to the matter or the aspect of it, as appropriate; or
(b) the clients are competing for the same objective5,
and the conditions below are met, namely that:
(i) all the clients have given informed consent, given or evidenced in writing, to you acting;
(ii) where appropriate, you put in place effective safeguards to protect your clients' confidential information; and
(iii) you are satisfied it is reasonable for you to act for all the clients.
These are linked from the above Code to give definitions and explanations from the SRA glossary:
1own interest conflict:
means 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
2conflict of interest:
means 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.
means the person for whom you act and, where the context permits, includes prospective and former clients
4substantially common interest:
means a situation where there is a clear common purpose between the clients and a strong consensus on how it is to be achieved
5competing for the same objective:
means any situation in which two or more clients are competing for an "objective" which, if attained by one client, will make that "objective" unattainable to the other client or clients, and "objective" means an asset, contract or business opportunity which two or more clients are seeking to acquire or recover through a liquidation (or some other form of insolvency process) or by means of an auction or tender process or a bid or offer, but not a public takeover
Can the right to self-represent be waived due to conflict of interest?
The existence of a current or prior relation does not forfeit the parties' right to litigate in pro per. The legal concept of conflict of interest has nothing to do with a party's decision to retain an attorney versus litigating in pro per.
From a legal standpoint, a conflict of interest arises where entity X is in a position favorable to advancing incentives which contravene or compromise the obligations X has toward entity Y. That favorable position most often, but not always, stems from information X obtained in his capacity of Y's agent.
Generally Y is unaware of, or presumed to reasonably rule out, those incentives. However, there are scenarios where Y might be perfectly aware of those incentives and yet be unable to preempt disparity from that conflict of interest. One scenario of that is a judge's refusal to recuse from a case where the judge's illegitimate incentives are likely to influence his rulings in that controversy, since the judge has an obligation to rule on the bases of legal or equitable grounds.