Do strings propagate through space time or do they make space time? In the beginning of string theory textbooks, strings are said to live in a background "target" space time. They then propagate through this space time. Strings also have a spin 2 ("graviton") mode, and can scatter off of each other. So in one sense, it seems like strings live "in" space time. Gravitational waves can then be thought of as coherent states of "string plane waves," constructed from asymptotic states.
But somehow, don't these strings also affect the space time they are in? It is often said that string theory has "no free parameters," but that different laws of physics (low energy effect QFTs) are given by different compactifications of extra dimensions. In some fuzzy sense, somehow these different compacifictions are minima of some potential energy (although I don't see how). What do these compactifications have to do the strings?
In other words, it seems as though a space time manifold is an input to your theory, and strings propagate in it. However it also seems like somehow the strings can select which space time they are in, because there are different possible compacifications of extra dimensions. So do strings live in space time, or do they "make" space time, somehow?
(I am a person who knows QFT but not much about string theory. I welcome any math used in an answer.)
 A: There are at least two different ways to think about the ontology of spacetime in string theory, with neither being any more correct than the other:


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*The non-linear $\sigma$ model view: One may view string theory as being a theory essentially defined by the stringy scattering amplitude as the sum over compact 2d manifolds on which certain conformal field theories live. All you need is a conformal field theory with non-anomalous Weyl symmetry on the manifold and the definition of the stringy amplitude, and you've got a string theory. Turns out that at least a certain subset of these theories - superconformal, $\mathcal{N}=2$ theories - are often naturally expressed in terms of an action of fields whose target space is a ten-dimensional manifold. 
In the general spirit of non-linear $\sigma$-models, we identify this target space with spacetime, and when you think about 2d manifolds, you'll realize you could claim that they are the worldsheets of 1d objects. But the formalism a priori doesn't require you to make this interpretation - we did emphatically not start from the assumption that we would be making a theory about strings floating through space to reach this point. Adherents of this ontology view both "strings" and "spacetime" as emergent aspects of "string theory", with the "fundamental" objects of the theory being abstract (S)CFTs living on donuts. 
Let me note that there is not a unique spacetime even for a fixed choice of CFTs, since there are mirror models, which are two different choices for the target space that nevertheless produce exactly the same string theory.

*The "strings floating through space" view: This approach is how string theory is usually taught: We start with the idea that we want to quantize the theory of a classical string floating through a spacetime, and jump through all sorts of ad hoc quantization procedures until we finally figure out that the only consistently quantizable theories are those with a vanishing Weyl anomaly, which in this approach is usually expressed in terms of an ordering constant arising due to operator ordering ambiguities during quantization. Here "spacetime" and "string" are fundamental objects, and the CFT aspect is something we learn along the way.
If we take the CFT viewpoint, then different compactifications are different choices for the CFT living on the worldsheet. If we take the other viewpoint, then different compactifications are different choices for the spacetime the string propagates through. No matter what, we will realize that string theory is intimately related to conformal field theory - and conformal field theories have a concept that's called deformation by marginal operators associated with their renormalization group flow. In an effective field theory POV (i.e. from the POV of the effective QFT on spacetime associated to a QFT), such a deformation corresponds to the variation of some vacuum expectation value of some field (or combination of fields). 
But, more importantly, it also corresponds to changing the shape of the target space of the CFT! That is, it directly corresponds to the so-called moduli of the target space. So a smooth variation in the CFT is associated to a variation in spacetime structure (but not a smooth one in the ordinary sense of classical geometry - the variation in moduli can cause the topology of the target space to change, and topology changes can't be smooth). If we understood the dynamics that governed the deformations of the CFT, then we would also understand how string compactifications are "selected", but AFAIK no universally accepted idea of what these dynamics should be exists among string theorists. Nevertheless, this seems strongly suggestive of a renormalization group flow-like mechanic ultimately determining the structure of spacetime.
For a lucid exposition of the "string theory as CFT on donuts" approach, I highly recommend Greene's "String Theory on Calabi-Yau Manifolds".
