On the other hand, Bayesians insist that probability is just an estimate of our beliefs.

A new paper tries to address the difference:

Forty some years ago David Lewis (1980) proposed a principle, dubbed the Principal Principle (PP), connecting rational credence and chance. A crude example that requires much refining is nevertheless helpful in conveying the intuitive idea. Imagine that you are observing a coin áipping experiment. Suppose that you learn -- for the nonce never mind how -- that the objective chance of Heads on the next flip is 1/2. The PP asserts that rationality demands that when you update your credence function on said information your degree of belief in Heads-on-the-next-áip should equal 1/2, and this is so regardless of other information you may have about the coin, such as that, of the 100 flips you have observed so far, 72 of the outcomes were Tails.I am not sure that any of this makes any sense.The large and ever expanding philosophical literature that has grown up around the PP exhibits a number of curious, disturbing, and sometimes jaw-dropping features.1 To begin, there is a failure to engage with the threshold issue of whether there is a legitimate subject matter to be investigated. Bruno de Finettiís (1990, p. x) bombastic pronouncement that "THERE IS NO PROBABILITY" was his way of asserting that there is no objective chance, only subjective or personal degrees of belief, and hence there is no need to try to build a bridge connecting credence to a mythical entity. Leaving doctrinaire subjectivism aside for the moment and assuming there is objective chance brings us to the next curious feature of the literature: the failure to engage with substantive theories of chance, despite the fact that various fundamental theories of modern physicsó in particular, quantum theoryó ostensibly speak of objective chance. Of course, as soon as one utters this complaint the de Finetti issue resurfaces since interpretive principles are needed to tease a theory of chance from a textbook on a theory of physics, and de Finettiís heirsó the self-styled quantum Bayesians (QBians)ó maintain that the probability statements that the quantum theory provide are to be given a personalistic interpretation.2

The only way I know to make rigorous sense out of probability is the Kolmogorov probability axioms.

I don't believe there is any such thing as objective probability. It has never been an essential part of quantum mechanics.

Quantum mechanics is about observables. Probabilities are not observable. Believing in physical/objective/propensity probability goes against the spirit of the theory.