Can any curve in 3D space be described by an intersection of two surfaces? If not, what assumptions I need to let it be true?
If this is too general, what if I restrict the scenarios to twice differentiable curves and surfaces? Suppose $\phi_i \in C^2 : \mathbb{R}^3 \rightarrow \mathbb{R}$. The two surfaces are represented by $\phi_1(x,y,z)=0$ and $\phi_2(x,y,z)=0$ respectively. Then the curve is the intersection of them: $L=\{(x,y,z) \;|\; \phi_1(x,y,z)=0, \phi_2(x,y,z)=0\}$.
Not sure if I understood your question. Do you want to recover surfaces from their intersections ?
Before getting into DG, simpler cases should be understood.
If a cone and a plane intersect we have the conic sections. When two second order surfaces intersect the orthogonal projections of intersection are of second degree.
A helicoid and cylinder intersect to produce a helix. If the 3d helix is given, I cannot imagine what other surface pair would produce the same helix.
The space curve of intersection is of simple description only in simple cases.
For example if an ellipse of given axes size is given then the intersecting surface pair can be cone/plane, cylinder/plane etc.
If a circle is given some rotationally symmetric surface pairs are possible eg, Cylinder/sphere, Cone /ellipsoid etc. on same axis as shown in rough hand sketch would arise, the pair is not unique.