Assuming that you mean that their secrets died with them except for what they told you, then you have a decision to make. On one hand, you must consider whether you continue to honor the request for secrecy. They made it while they were living, perhaps in order to avoid conflict, embarrassment, social awkwardness, shame or social stigma, etc. You have to consider the nature of their secret and why they didn't want others to know at the time. Will anything positive potentially come of divulging that secret now? You also have to consider whether those reasons are still valid, now that your loved one is no longer here.
The other alternative is that in the aftermath of their death their relatives and friends are searching for any answers that may potentially give more texture to the deceased's life and their passing. This is especially the case if they died by their own hand or by another's.
If divulging their secret has the potential to hurt the living and complicate others' grief -- information about extramarital affairs, previously undisclosed sexual identity, disease status, paternity, criminal or abusive behavior -- consider discussing first with a therapist to work out the following:
1) whether you should share the information
2) how much information you should share
3) with whom
4) in what manner/format (letter/phone/meeting) and what setting
5) the timing
6) what your own motivations are for wanting to do so.
I'm sorry for your loss. Some secrets are meant to remain secrets whereas others are not, especially after death.