= Solution
The uniform charge density is
$$
\rho_q=\frac{3Q}{4\pi R^3}.
$$
Spherical symmetry and <Gauss's law> give
$$
\boxed{
\mathbf E(\mathbf r)=
\begin{cases}
\displaystyle
\frac{Q}{4\pi\varepsilon_0R^3}\,\mathbf r,
&|\mathbf r|<R,\\[6pt]
\displaystyle
\frac{Q}{4\pi\varepsilon_0}
\frac{\mathbf r}{|\mathbf r|^3},
&|\mathbf r|>R.
\end{cases}}
$$
Take the first centre as the origin, so the second centre is at $\mathbf d$. In the overlap, each point lies inside both spheres. Superposition gives
$$
\mathbf E_{\rm total}
=\frac{Q}{4\pi\varepsilon_0R^3}\mathbf r
-\frac{Q}{4\pi\varepsilon_0R^3}(\mathbf r-\mathbf d)
=\boxed{\frac{Q}{4\pi\varepsilon_0R^3}\mathbf d}.
$$
The field is therefore constant throughout the overlap region.
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