I can't manage to find a way to enter overlapping notes in Sibelius 7.
By overlapping notes I mean notes from other voices that have a longer duration than others which are placed at the very same beat.
In Sibelius, you can use up to four separate voices on a single staff. The stems on notes for voices 1 and 3 point upward, while those for voices 2 and 4 point downward. The four voices are accessible via the numbered buttons labeled 1–4 and "All" at the bottom of the keypad menu. The "all" button selects all voices which are currently in use in the selected passage of music, and whatever change you make will affect all voices.
You should be able to enter music for the upper voice normally, then press the button labeled "2" to select the second voice and enter the music for the lower voice. The durations of the two voices will be completely separate, allowing for "overlapping" notes.
Just make sure to be aware of which voice you are editing at all times so that you don't accidentally change the wrong voice! Of course, even if this does happen, you can always select the music, right click, hover over "voice," and then select which voice you want to move the selected passage to.