There are a bunch of `Region` functions that just showed up in 10.  This uses `RegionDistance[]` to make a function that computes the shortest distance from a point to a region.  The generated function runs faster than just checking all the circles.  Though the creation of the function in the first place has to look at all the circles.  So this still runs in $\mathrm O(n^2)$ time, but at least the inner `While` should be faster when there are already a lot of circles and not a lot of places to put a new one.

    distinct[n_,r_]:=Module[{d,f,p},
      d={Circle[RandomReal[{-1,1},2],r]};
      Do[
        f=RegionDistance[RegionUnion@@d];
        While[p=RandomReal[{-1,1},2];f[p]<r];
        d=Append[d,Circle[p,r]],
        {n-1}];
      d
    ]

Example:

    distinct[75, 0.1] // Graphics

![75 non-overlapping circles][1]

The following runs quite a bit faster, generating a `Nearest` function for a large set of circles, and using that one function to eliminate many overlaps in a single pass.

    sweep[pts_,r_]:=Module[{p,f,c},
      p=pts;
      While[
        f=Nearest[p->Automatic];
        c=f[p,{2,r}];
        Last[Dimensions[c]]!=1,
        p=Pick[p,First[#]<=Last[#]&/@c];
      ];
      p
    ]

    distinct2[n_,r_]:=Module[{p},
      p={};
      While[
        p=Join[p,RandomReal[{-1,1},{n,2}]];
        p=sweep[p,2r];
        Length[p]<n
      ];
      Circle[#,r]&/@Take[p,n]
    ]

Measuring the performance (your mileage may vary):

    Timing[distinct[75,0.1];]

>     {3.021209,Null}

    Timing[distinct2[75,0.1];]

>     {0.077105,Null}

A problem with both of these algorithms is that there is no assurance that they will halt.  You can ask for too many circles, and end up with the `While` loops trying to add circles forever when there is no place to put even one.  If this is supposed to be part of some larger automated process, then you need to add a check for no progress after so many tries in order to give up.  E.g.:

    distinct3[n_,r_,maxiter_]:=Module[{p,m,k},
      p={};
      m=0;
      k=maxiter;
      While[
        p=Join[p,RandomReal[{-1,1},{n,2}]];
        p=sweep[p,2r];
        k=If[Length[p]>m,maxiter,k-1];
        m=Length[p];
        m<n&&k>0
      ];
      Circle[#,r]&/@If[Length[p]>n,Take[p,n],p]
    ]

Then:

    distinct3[100,0.1,1000]//Length
>     80

    distinct3[100,0.1,1000]//Length
>     78

As pointed out by Ernst Stelzer, this works in 3D just as well with two small changes:

    distinct3D[n_,r_,maxiter_]:=Module[{p,m,k},
      p={};
      m=0;
      k=maxiter;
      While[
        p=Join[p,RandomReal[{-1,1},{n,3}]];
        p=sweep[p,2r];
        k=If[Length[p]>m,maxiter,k-1];
        m=Length[p];
        m<n&&k>0
      ];
      Sphere[#,r]&/@If[Length[p]>n,Take[p,n],p]
    ]

Then:

    distinct3D[150, 0.2, 100]//Graphics3D

![118 spheres of radius 0.2 packed into 2x2x2 box][2]

That is 118 spheres.

  [1]: https://i.sstatic.net/9uq4a.png
  [2]: https://i.sstatic.net/yxnoY.png