In both cases, use of their is likely to convey the impression that they are representing a larger group (thus, Andy is giving the results obtained from the work of several people). If the results are the work of Andy alone, and assuming that Andy is male, then his will not be persecuted by the gender-police, and will avoid the ambiguity around who obtained the results. Similarly, if Catherine alone obtained the results and is female, then her is appropriate.
And sharing their results sounds unnecessarily PC-ish. A word such as explaining, exposing, discussing, detailing, describing, ... would remedy this.
Edit In response to the comment by the OP and Andy Schweig, how about yy's results [...] will be described by their author (author may not be the best term).