Imagine the following gedankenexperiment. Observer Alice is right here on Earth. Observer Bob is at say Alpha Centauri. A pair of maximally entangled qubits is formed with one qubit handed over to Observer Alice and the other sealed in a box on Pluto shielded from decoherence. It takes a few years for any signal to reach Observer Bob. Observer Alice measures her qubit and "collapses" it. To her, the pair of qubits is no longer entangled. However, for a few years, will Observer Bob think there is an entanglement? We have to be careful here. Once the qubit on Earth has been measured by Earthlings, the entanglement is "shared". By the well known monogamy of entanglement theorem, any correlation between both qubits by themselves will have become classical. However, Observer Bob, who has a very good working knowledge of quantum mechanics and relativity, knows that for a few hours, until light signals can reach Pluto, the system consisting of an expanding bubble around Earth and the Pluto qubit are in an entangled state. Alice, who lives inside this bubble, will beg to differ. Is quantum entanglement an objective or subjective property?
Quantum entanglement is an objective property.
Naughty Troublemaker described a very interesting situation that make quantum entanglement looks like a subjective property. However, even in such a situation, quantum entanglement is still an objective property.
After Observer Alice measures her qubit and "collapses" it, the quantum entanglement is still there, but now the Bob's qubit is entangled with a combined system formed by Alice and her qubit. So before the measurement, Bob's qubit only entangles with Alice's qubit. After Alice measures her qubit, Bob's qubit entangles with a combined system formed by Alice and her qubit.
Entanglement is objective. This is easily seen in a simple example from nuclear magnetic resonance (NMR). Start with two spins governed by an 'AB' static Hamilitonian, that is, withscalar coupling comparable in magnitude to the chemical shift difference. (Look this up in Abragam's book if you are not familiar.)
It is impossible to make a one-quantum transition between eigenstates without disturbing both spins -- it is impossible to perturb one without perturbing the other.
In other terms, calculate the evolution of the density matrix following a nutation of both spins by pi/2. The two-spin density matrix is not (repeat not) factorizable into the Kronecker product two one-spin density matrices.
The correct (machine observable) free induction signal cannot be computed without the entangled two-spin density matrix. Nothing philosophical or speculative above this.
Therefore, Engtanglement is Objective.
As the OP implicitly suggests, quantum entanglement is a subjective property. It is an aspect of a subject's knowledge about some subsystems of the physical system one considers, namely some knowledge about the correlations.
Just like the "collapse" isn't an objective process but a change in the subjective knowledge about the observables, the detailed properties of the wave function – which is itself a collection of numbers encoding the subjective knowledge – including the entanglement between subsystems is subjective. The moment when the wave function "collapses" depends not only on the reference frame in the special relativistic sense; it also depends on whether or not an observer (a large enough object) is willing to associate a scheme of mutually complementary histories to some events affecting a physical system.
Much like in many other places in physics, this fundamental subjectivity or observer dependence leads to no contradictions and in most cases, it may be approximated by "objective properties". However, it's not possible to do so exactly. Fundamentally speaking, the wave function and all of its properties and events such as the entanglement and its "collapse" are just subjective entities or processes.
Well, for entanglements which are actually measurable in practice, there is practically an intersubjective agreement that entanglement exists. The OP's example involves a sort of "entanglement" which can only be measured by reflecting all the outgoing signals from Earth and a Loschmidt reversal. Impractical. In fact, whether such examples are entangled or not isn't empirical. The only reason why physicists doubling as armchair metaphysicians think there is entanglement is because their theory (relativistic quantum field theory) tells them that according to their theoretical framework, there ought to be entanglement. Purely rationalistic stuff. If you can't measure entanglement physically, it's not really entanglement. It's "head in the clouds" theoretical speculation.
If I hand you an ensemble of N photon pairs, such that each pair is entangled (but no pair is entangled with any other pair), you can detect their correlation with fidelity 1 - 2^-N. Another way of saying this is that, on measurement (2 measurements per pair) they will all be either the same (00 or 11) or all be different (01 or 10), up to system noise.
If I then hand you an ensemble of N uncorrelated photon pairs, you will measure randomness - i.e. you can directly observe the lack of correlation. Therefore entangled pairs look different to unentangled pairs simply by making two measurements per pair on an ensemble of size N pairs.
If N=1, you can only distinguish entangled from unentangled in the special case that you know a) the sign of the entanglement correlation, and b) measurement on the observed pair shows the opposite correlation. In this case entanglement is absent.