Praise be to Allah.
Firstly: the person who is doing sajdat al-tilaawah should say Takbeer when going down, because of the hadeeth narrated by Abu Dawood in his Sunan from Ibn ‘Umar (may Allah be pleased with them both) who said: “The Messenger of Allah (peace and blessings of Allah be upon him) used to recite Quran to us, and if he reached a verse where a sajdah was prescribed, he would say Takbeer and prostrate and we would do likewise.”
(Reported by Ahmad, 2/17; al-Bukhaari, 2/33,34; Muslim, 1/405, no. 575). He should not say Takbeer when coming up from the sujood, because there is no report from the Prophet (peace and blessings of Allah be upon him) to indicate that, and because sajdat al-tilaawah is an act of worship, and acts of worship are to be done in accordance with what is prescribed in sharee’ah. What was reported from the Prophet (peace and blessings of Allah be upon him) was Takbeer when going down into sujood al-tilaawah, not when coming up. But if a person is doing sajdat al-tilaawah when praying, then he should say takbeer both when going down and when coming up, because of the general meaning of the saheeh ahaadeeth which describe the prayer of the Prophet (peace and blessings of Allah be upon him) and state that he said Takbeer each time he went down into sujood or came up.
Secondly: one should not recite al-Tashahhud or say salaam after doing sajdat al-tilaawah, because there is no proof that the Prophet (peace and blessings of Allah be upon him) did that. Sajdat al-tilaawah is an act of worship and as such should be done as prescribed as in sharee’ah and we cannot make qiyaas (analogy) comparing it to the tashahhud and salaam in prayer.
And Allah is the source of strength. May Allah bless our Prophet Muhammad and his family and companions.