Consider a statement of the form "based on knowledge and common sense and estimating the probabilities of alternative hypotheses, I believe that X". Or, perhaps, "based on Bayesian reasoning I have come to the conclusion X". How much, when an interlocutor of yours uses such a statement, should it affect your credence in X?

I claim that the answer is "negligibly", in most cases. (I discuss exceptions below.)

The most appropriate way of conducting an argument is in most cases to simply state your belief X. If you are going to adduce evidence and argument in support of your belief X, simply state these. If you've derived probability estimates, simply provide them, perhaps with the method of derivation. The "meta" observation by itself has no place in an argument.

(Isn't that obvious, you might ask? Not to everyone, as this statement form turns out to be actually used in discussions here. So, in the interest of contributing even a little to a possible theory of argumentation, I develop some further observations on this pattern.)

The assertion "my opinions on X are soundly arrived at" has an equivalent structure to saying "I am a truthful person": it cannot be verified by an interlocutor, other than (in the latter case) by looking at what statements you utter and independently verifying they are truthful, or (in the first) by looking at what arguments and evidence you adduce in support of your opinions on topic X, and independently assessing whether the evidence is credible and the arguments are individually sound.

Claiming to be a truthful person doesn't mean much. I know parents who are constantly telling their kids "oh, you shouldn't lie, it's bad", but who actually lie to their kids quite often. My own approach has been to tell my kids the truth (and to answer pretty much all their questions), while almost never mentioning "truth" as a moral topic. Empirically, I observe that my kids are growing to be truthful and trustworthy persons, more so than many of e.g. my neighbours' kids.

Claiming that you use sound methods in arriving at a conclusion of which you want to convince your interlocutor similarly has little content. It is the sort of thing that can only be assessed by looking at your justifications for belief. The claim may (if stated in an authoritative tone) actually manage to nudge your interlocutor a little in the direction of accepting your conclusions. On reflection, it shouldn't, and your argumentative skills would benefit from refusing to accept such statements as justification.

Rather than say "there is evidence", point to evidence. Rather than say, "it can be argued that", simply argue it. If there are allowable exceptions, they concern cases where you say something in addition to the bare assertion of soundness: for instance if you say "there is lots of evidence, readily obtainable from everyday sources". Your claim is above and beyond a claim of soundness.

You should be prepared, if challenged, to back up your observation about the abundance and availability of the evidence. If you say of something that it's "obvious", it had better be obvious.

Sometimes too, the use of a given methodology to arrive at a belief X is surprising information in and of itself. It makes a difference to an interlocutor to know that your belief has that origin. It is hard to think of examples where it could possibly make a difference to say that in very vague and general terms, but I can think of cases where the use of a specific method is new and surprising. For instance, if you say "I believe in X and Bayesian reasoning (as opposed to other methodologies) demonstrates X in a particularly convincing manner". In this case the claim is not only about X, it is also a reasonably interesting claim about the method itself. (Which claim might or might not stand up to examination -  cf. the ongoing argument about frequentism vs. bayesianism.)

These exceptions aside, sound argumentation shares one precept with fiction: show, don't tell.