I do not know, what you mean by "qualitative" understanding. Perhaps you mean intuition, and of course, we don't have an intuition about entanglement, because in everyday life we don't have experience with such phenomena. The reason is that we are surrounded by macroscopic systems, with an overwhelming number of coupled microscopic degrees of freedom. Fortunately, our senses coarse-grain over all the many unimportant microscopic details, and the relevant macroscopic observables behave to a utmost high accuracy according to the laws of classical physics. That's why we have found the classical description of nature before we knew about the quantum nature behind. The only way we can understand quantum theory is through mathematics and the "mapping" between mathematical abstract structures (Hilbert-space vectors, statistical operators, and operators representing observables, etc.) to the real world (Born's probability rule for the interpretation of quantum mechanical states, spectral theory of operators).
If it comes to non-locality one has to be a bit careful, what one means. The most comprehensive quantum theory we have today is relativistic quantum field theory, which is by construction a theory of local interactions. All causal actions are thus described by local interactions. "Local" means here that we describe systems of elementary particle with a set of field operators and a Hamiltonian that is derived from the spatial integral over polynomials of field operators at the same space-time point. The field operators also fulfil local transformation laws under proper orthochronous Lorentz transformations.
Using such a description of a quantum system implies the socalled "linked-cluster principle", which states that experiments on systems that are very far away from each other are stochastically independent, i.e., the probabilities for different local subsystems at far distances factorize.
On the other hand, quantum states can describe non-local correlations, and that's what's commonly discussed when it comes to entanglement. E.g., nowadays quantum opticians can easily produce entangled two-photon states. The important point is that these are real two-photon states, i.e., a Fock state with a precise photon number of two. The entanglement of the photons in such a state, usually produced with help of parametric down conversion by shooting a laser through a birefrigerent crystal, are entangled with respect to their polarization state. This we cannot describe with everyday language, and we have to go to the level of mathematics. The state, I have in mind is of the form
|\psi \rangle=\frac{1}{\sqrt{2}} [|HV \rangle-|V H \rangle].
I've noted only the polarization part of the single-photon states, and I use the usual shorthand for tensor products |H V \rangle:=|H \rangle \otimes |V \rangle.
In principle, for the following discussion, I'd have to also note the spatial part of these states, but that's cumbersome, and I hope I can make clear my point of view in this somewhat simplifying notation.
The point is that this photon state is prepared at the very beginning using a local device, namely the crystal for the parametric down conversion. Then the two photons propagate without further interactions, and after some time the spatial probability distribution for measuring one of the photons is peaked at very far distant positions (note that I don't talk about positions of photons, which cannot be defined in a simple way, but that's not so much an issue here). So Alice and Bob put far distant photo detectors with polarization filters in the direction of these two spots of high probability to detect a photon. Each of them measures single photons.
First of all we may ask about the probability that, say, Alice detects a photon with a certain polarization, if her polarization filter is directed in horizontal direction. This is described by the socalled reduced statistical operator for the one-photon subsystem. With the above given state, Alice thus describes the state of her one photon as
\hat{\rho}_{A}=\text{Tr}_2 |\psi \rangle=\frac{1}{2} \hat{1}=1/2 (|H \rangle \langle H | + |V \rangle \langle V|).
This means, she has a totally unknown polarization state. If she samples very many photons, using all angles of her polarization filter, she'll come to the conclusion that she simply has an unpolarized photon source. The same holds true for Bob. Both cannot conclude that their photons come from a single source of entangled photon pairs with their local measurements alone.
Now they can also measure the time of their photon registration. Now suppose that Alice points her polarizer in H direction and Bob his in V direction. Then our entangled state is such that whenever Alice registers a photon (which is, of course, only in 50% of all produced photon pairs) also Bob must register his photon (assuming detectors with 100% efficiencies) too. That means there is a 100% correlation between Alice's and Bob's polarization state of their photons, although the polarization state of each of their photons is maximally random (unpolarized photons!).
Now, this experiment has been done such that the registration events were spacelike. According to relativity thus there cannot be any causal effect of Alice's photon detection on Bob's and vice versa. Thus, the non-local correlation cannot have been caused by Alice's or Bob's measurement, and the above description of the quantum theoretical analysis should make it very clear that such an assumption is not necessary to make at all. In this minimal interpretation, there is no necessity for a "collapse of state" or any other mysterical "spooky action at a distance". This is only due to the collapse interpretation of the Copenhagen and related schools of quantum theory. If one sticks to the minimal necessary assumptions, i.e., Born's probability rule as the interpretation of the quantum-theoretical states, no such assumptions have to be made, and there is no contradiction between quantum dynamics and Einstein causality, according to which no signals can propagate faster than with the speed of light. Thus also the criticism by Einstein, Podolsky, and Rosen against quantum theory becomes immaterial. According to the Minimal Interpretation, of course, nature is inherently probabilistic, i.e., indeterministic. An observable has only a determined value if the system has been prepared in an eigenstate of the corresponding operator, describing this observable. Otherwise this observable doesn't have a determined value, and you can only give probabilities for the outcome of measurements of this observable. That's it.
Whether you consider this as a "complete description" of nature or not, is your own belief. It has nothing to do with nature which behaves as it behaves. Physics states as precisely as it can facts about phenomena in nature and tries to describe this behavior with mathematical theories. In this way you can make predictions about the probabilities of measurements, given a previous preparation of the system. Experiments have then to use ensembles of independently prepared systems and get, within the statistics of the experiment, the probabilities for measuring the values of the observables of such prepared ensembles and compare it with the the quantum-theoretically predicted ones. So far, quantum theory has survived all tests to a very high precision. Also the non-classical features of entanglement with their non-local correlations that cannot be described by classical local hidden-variable theories (Bell's and related inequalities), have been confirmed with a huge significance. In this sense we must come to the conclusion that quantum theory is indeed a complete description of nature and from this we must conclude that nature is inherently non-deterministic.
