Is it possible to make a state tomography of an entangled state only by measurements on subsystems? As far as I can understand, to make a state tomography of a composite system, a properly selected set of joint measurements should be carried out on the composite system to estimate the density matrix of the system. 
My question is, for a general entangled state, or just a two-partite entangled system, is it possible to achieve a state tomography by only carrying measurements on its subsystems? 
Due to the general idea that the global information of entangled states can not be obtained by descriptions of subsystems, the answer seems to be NO. But I do not know if there is a definite conclusion on this or some other related results.  For example, I heard of 'local cloning' of entangled states. If cloning is also based on information of the composite system, how the 'local cloning' can obtain enough information to achieve a cloning of the entangled system? Is there any connection between the 'local tomography' and 'local cloning' of entangled states?
 A: It is possible to do tomography by only doing measurements on the subsystems, as long as one is able to correlate the measurement outcomes, this is, we have to be able to measure expectation values (w.l.o.g. I restrict to a bipartite state)
$$
\langle A\otimes B\rangle = \mathrm{tr}(\rho(A\otimes B))\ .
$$
The reason is that in order to do tomography, it is sufficient to measure $\mathrm{tr}(\rho X)$ for a set of (hermitian) operators $X$ which generate the full space of (hermitian) matrices.
To understand how this works, note first that  $\mathrm{tr}(X^\dagger Y)$ defines a scalar product (=a Hilbert space structure) on the space of matrices, and for a Hilbert space, a vector is completely determined by its scalar product with any basis of vectors. On the other hand, given two Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$ with bases $\vec c_i$ and $\vec d_j$, a basis of $\mathcal H_1\otimes \mathcal H_2$ is given by $\vec c_i\otimes \vec d_j$.
Thus, all you have to do is to pick hermitian matrices $C_i$ and $D_j$ which span the full space of hermitian matrices, and measure all expectation values (=determine the scalar products)
$$
\langle C_i\otimes D_j\rangle=\mathrm{tr}(\rho(C_i\otimes D_j))\ ,
$$
which is enough information to reconstruct $\rho$.
For instance, in the case of two qubits, it is sufficient to choose $C_i$ and $D_j$ the Pauli matrices and the identity, and thus, it is sufficient to measure
$$
\langle \sigma_i\otimes \sigma_j\rangle
$$
with $j=0,1,2,3$.
It is easy to see that this argument generalizes to the multipartite setting.

EDIT (following a comment by the OP):
If one wants to restricts to measurements on one part only, i.e., 
$\langle M_i\otimes Id \rangle$ and $\langle Id\otimes N_j\rangle$, tomography is not possible, since these expectation values only depend on the individual reduced density matrices. A simple counterexample is given by any two maximally entangled states, such as 
$$
\vert\psi^{\pm}\rangle = \frac{\vert 00\rangle \pm \vert 11 \rangle}{\sqrt{2}}\ ,
$$
since for any maximally entangled states, the reduced density operators for both parties are the maximally mixed state.
