How do we know that two quantum states are entangled?

When two particles are entangled, is there a way of determining that a particle is entangled if one does not have access to both particles?

is there a way of determining that a particle is entangled if one does not have access to both particles?

No, you need access to both, and you need to be able to collect statistics through repeated measurements on an ensemble of such setups that have all been prepared in the same way. Even then, it can be tough. This is called the quantum separability problem, and it's known to be intractable in a certain technical sense.

The much used term "entanglement" means that there exists a quantum mechanical wavefunction describing the particles. This means that one knows the particles participating in the interaction and the initial conditions. Then the complex conjugate squared of the wavefunction will predict the probability distribution of the decay or interaction. This means in general that the experiment has to be repeated enough times so that the statistical distributions give an accurate measurement, as the other answer states.

There is an exception of the need for statistics, when one measures conserved quantum numbers, that have to be conserved in an interaction. Given the existence of two particles, the measurements of the quantum numbers of one particle gives information on the quantum numbers of the other which is not measured.

A simple example is the $$π0 -> γ γ$$ meson. If the input four vector is known, i.e. that the mass is that of the $$π0$$ then measuring the spin of one of the $$γ$$ informs us that the entangled one that was not detected has the negative spin, because the $$π0$$ has spin zero, and angular momentum is a conserved quantum number.

So it is by use of the mathematical model of the interaction that in particular cases the state of the unmeasured particle is determined also.

Having access to both particles won't help. Given a measurement (on one particle or on the entire system), and given an outcome of that measurement, there is always some non-entangled state that yields that outcome with nonzero probability. (In fact, there's always a non-entangled state that yields that outcome with certainty!). So no measurement can reveal that the inital state was entangled, and you can never know that a system is entangled unless you know its history.