It would be easier to give you some help if (1) you indicate why you only want to use the data locations with in a polygon to interpolate within that polygon (2) you clarify whether you are wanting to generate interpolated values at other points in the same polygon or possibly to estimate an average value for each polygon. It is almost certain that you are going to introduce some "edge effects" that will be difficult to quantify.

What interpolation algorithm are you proposing to use? Some algorithms such as kriging depend on statistical assumptions that may not match with the polygon decomposition.

What are you going to use the interpolated values for?

Note that simple minded interpolation methods such as "natural neighbor" and Inverse distance weighting are not as reliable because they do not incorporate as much information.

In a similar situation (~2000 repetitions) I made an interpolation for each polygon separately, but with using a script for automation of a process.

I don't know the answer, but I have a question. Will it affect the interpolation if you interpolate and cookie cut with the polygon layer? For what kind of application do these differences matter? Just for information ...

Interpolation it´s just a kind of dimensional transformation from points to surface, then if you need interpolate from boundaries, first you need apply an dimensional tranformation from bonudaries to points.

There is two possibilities:

A. If you have just one point then apply the next process:

1, Relativize the data to be interpolated, for surface or for population. YOU CANT INTERPOLATE ABSOLUTES. 2, Join the dta of the point to poligon. 3, Get centroid from poligons with the associate data you need interpolate. 4, Use a method not contradictory with polygons, like TIN. DON´T USE GRAVITATIONAL (IDW) what have not the same logic that boundaries.

B. If you have more than one point for each poligon then you have at least two ways,

a.

1. Apply spatial join from points to poligon, 2.repeat steps 1 to 4 in the last method

b.

1. Construct a spatial data matrix (grid) (In Vector menu) 2, Apply spatial joint from point to matriz, 3, interpolate with natural neighbor or with TIN. Look that all the data in a grid, has been relativized for its own nature.

It would be easier to give you some help if (1) you indicate why you only want to use the data locations with in a polygon to interpolate within that polygon (2) you clarify whether you are wanting to generate interpolated values at other points in the same polygon or possibly to estimate an average value for each polygon. It is almost certain that you are going to introduce some "edge effects" that will be difficult to quantify.

What interpolation algorithm are you proposing to use? Some algorithms such as kriging depend on statistical assumptions that may not match with the polygon decomposition.

What are you going to use the interpolated values for?

Note that simple minded interpolation methods such as "natural neighbor" and Inverse distance weighting are not as reliable because they do not incorporate as much information.

(1)If you try to interpolate in different polygons far away to each other, Petr has given you the answer. 

(2)if you try to interpolate in polygons adjacent to each other, it is impossible to get a continuous surface using IDW or KRIGING in each polygon. Spline interpolation with smoothing would do the job. I don't see QGIS has the tool.

http://gracilis.carleton.ca/CUOSGwiki/index.php/Creating_IDW_and_Spline_Interpolation_Maps_Using_QGIS#Creating_Clipping_Polygon_and_Clipping_Interpolation_Surface

Has anyone tried areal interpolation in QGIS?

I tried intersect, and found each time the intersect function produces a different number of polygons. Why is that?