Does scripture say to maintain secrecy in spiritual practices? Are we not recommended to speak about our Ishta Deva, spiritual practices or devotion?
Yes, certain amount of secrecy is to be maintained when it comes to spiritual practices.
For example, certain religious activities, like charity, and certain spiritual activities, like austerity, must always be kept secret.
The Daksha Smriti (DS), mentions about 9 such acts which a householder must always keep as secret.
Iniquitous deeds are, again, [the following : ] Falsehood, knowing another's wife, taking forbidden food, knowing a woman who should not be known, drinking what should not be drunk, theft, committing injury, doing works not sanctioned in the S'ruti, transgression of a friend's duty, these are nine improper deeds. One should avoid them all.
Longevity, wealth, weakness of a house, counsel, sexual intercourse, medicine, austerity, charity, and honour, these nine should be carefully kept secret.
DS, Chapter 3 verses.
And, as regards Mantra DikshA, the very first instruction that a Guru gives to his disciple (after the DikshA is complete) is " Do not ever reveal the Mantra to anyone". Every disciple gets this instruction for sure whether he gets some others or not.
So, you can say to anyone " Hey, Sri X is my Guru" but you should never say to anyone " Hey, i am being initiated into Mantra M".
This order is actually Lord Shiva's following words from the KulArnava Tantram (KT):
Gurum prakAshayed dhimAn mantram yatnen gopayet |
AprakAsha prakAshAbhyAm nasyatah sampadAyushi ||
The Guru must always be revealed by the wise, but the mantra must carefully be kept secret. By not doing these two acts- that is, by not revealing the Guru and by not concealing the mantra, decrease, respectively, [the disciple's] wealth and longevity.
On similar lines, Lord Shiva further says to Mother Goddess:
Sriguroh pAdukAm mudrAm mulamantram swapAdukAm |
ShishyAdanyasya deveshi na vadet yashya kasyachit ||
The Teacher's PAdukA mantra, his mudrA, one's own mulamantra and pAdukA mantras can only be disclosed to one's own disciple. It should never be told to anybody else.
So, there is definitely a need for maintaining secrecy.
Further, one should also not reveal the deeper secrets embedded in his own Sampradaya scriptures to others.
SwashAstroktam rahsyAdyam na vadet yasya kasyachit |
Yadi bruyAt sa samayAcchyutta eva na samsayah ||
The disciple must not disclose the mysteries of [his own Sampradaya] scriptures to anyone [without discrimination]. One who does so, without doubt, falls from his conduct.
Now, these are some general advice that are applicable to each and everyone, irrespective of which sect he may belong to.
And, for the Kaulas (not exactly the same, but loosely- those who belong to the ShaktA sect), there is clear instruction given by Lord Shiva, in possibly every ShAkta Agamas, that one should not reveal that he is actually a Kaula to others. That's why he says:
Antah kaulo vahih shaivo janamaddhye tu vaishnavah |
Kaulam sugopayed devi nArikela phalAmvuvat ||
O Goddess, from inside (in heart) Kaula, from outside (in behavior) Shaiva, among masses, behaving like a Vaishnava, a Kaula must always conceal the Dharma of Kaula, just like that water which remains hidden in the coconut.
In MAtrikA Bheda Tantram, Lord Shiva discusses several Doshas or afflictions that negatively affects the Mantra DikshA, and one among them is called the "Mantra Kathana Dosha"- or the affliction that arises when one reveals his Guru Mantra to others.
Lord Shiva says that this offence is quite serious and can eventually lead to the disciple's death unless the prescribed prAyaschittas are performed by him.
Can one reveal who his Ishta DevatA (favorite Deity) is?
Not sure on this one. Probably Yes. At least, till now, i have not read anything that says No.
Updating the answer from Hatha Yoga Pradipika (HYP), according to which, Yogic practices are to be kept secret as well:
Hathavidyā param gopyā yoginā siddhimichchatā |
bhavedvīryavatī guptā nirvīryā tu prakāśitā ||
A Yogî desirous of success should keep the knowledge of Hatha Yoga secret; for it becomes potent by concealing, and impotent by exposing. (HYP 1.11)
Karma shatkamidam ghopyam ghataśodhanakārakam |
Vichitraghunasandhāya pūjyate yoghipungavaih ||
These six kinds of actions which cleanse the body should be kept secret. They produce extraordinary attributes and are performed with earnestness by the best of Yogîs.
There are actually many such verses in the text to that effect.
Mochanam netraroghānām tandādrīnām kapātakam | Yatnatastrātakam ghopyam yathā hātakapetakam ||
Trâtaka destroys the eye diseases and removes sloth, etc. It should be kept secret very carefully, like a box of jewellery.
So, even for Yogic practices like meditation secrecy to be mentioned.