I have done some tests.
See attached EPS example.odp which has three images. Top left is a .eps file of "black trees" which has no transparency. Bottom left is a PNG file with transparency surrounding the red square. Top right is a .EPS file created from the red square with transparency, so has transparency. The PDF is then created from the .odp file.

- Clipboard01.png (9.97 KiB) Viewed 5906 times
Conclusions seem to be
1 A .eps file
without transparency displays properly on screen. Nothing appears in the PDF.
2 A .eps file
with transparency does not display on screen - not even the solid component. Nothing appears in the PDF.
3 A .png file with (or without) transparency works as expected and items can be seen through the transparency. It exports correctly to the PDF. If I copy the image out of the PDF the transparent part which shows as white is copied as black.
Hence my suggestion is to convert your .eps files to PNG or JPG as appropriate and insert them as PNG or JPG files.