Is the pronunciation of "secret" /ˈsiːkrət/ or /ˈsiːkrɪt/?

It depends on, but is not limited to, the accent of the speaker. English dictionaries provide only the two standard pronunciations: RP and GenAm. In terms of pronunciation, don't take for granted everything you see in dictionaries, because they employ simplified transcription. Additionaly, words in dictionaries are pronounced in isolation.

To give weight to my arguments, the GenAm transcription of the word "man" doesn't take into account the /æ/ tensing process, and yet you can clearly hear the difference between the RP and GenAm pronunciations of the word man. Yet, both transcriptions have been transcriped the same: man Someone unaware of this process might incorrectly assume that in fact both pronunciations are the same.

Another example includes the flap T (the alveolar tap). Americans flap their Ts, and yet the flap T isn't transcribed, nor pronounced as /ɾ/. Why? Beats me.

Now, to answer your question, you would be better advised to check on the Internet the 3 varieties of unstressed vowels: schwa, schwi, and schwu. This article should shed some light on the issue and probably answer your question at least to some extent: https://en.wikipedia.org/wiki/Stress_and_vowel_reduction_in_English#Reduced_vowels_in_the_close_unrounded_area