We know the basic lexical meaning of τίθημι, and we know ultimately Jesus referred to his sacrificial death on the cross. But, in Jesus' figurative statements like this in John, they have an everyday life meaning for what Jesus used for the illustration as well as the Jesus’ ultimate meaning. The translations give Jesus’ ultimate means, but that meaning seems too extreme for what his listeners would expect from a good shepherd.
Ἐγώ εἰμι ὁ ποιμὴν ὁ καλός. ὁ ποιμὴν ὁ καλὸς τὴν ψυχὴν αὐτοῦ τίθησιν ὑπὲρ τῶν προβάτων· (John 10:11, NA28)
I am the good shepherd. The good shepherd lays down his life for the sheep. (John 10:11, ESV)
The Bible Society of Israel and Franz Delitzsch translated with נתן (give); Delitzsch using imperfect, 3rd masculine singular and the Bible Society using the participle masculine singular. However, in Matt 20:28 and Mark 10:45 when Jesus said “to give his life,” (δοῦναι τὴν ψυχὴν αὐτοῦ) he qualified it with “as a ransom” (λύτρον).
..even as the Son of Man came not to be served but to serve, and to give his life as a ransom for many.” (Matt 20:28, ESV)
For even the Son of Man came not to be served but to serve, and to give his life as a ransom for many.” (Mark 10:45, ESV)
The Peshitta translates τίθησιν with ܣܳܐܷܡ piel active participle masculine singular of sm put, place essentially the same root as שים, or BDB spells the root שום. This root can have the meaning to risk one’s life but with this meaning has the phrase “into one’s palm or hollow of one’s hand.”
take one’s life (נֶפֶשׁ) into one’s palm (i.e. risk it) Ju 12:3; 1 S 19:5; 28:21; Jb 13:14; -- Brown, F., Driver, S. R., & Briggs, C. A. (1977). In Enhanced Brown-Driver-Briggs Hebrew and English Lexicon (p. 963). Clarendon Press.
In particular look at the passage.
And Jonathan spoke well of David to Saul his father and said to him, “Let not the king sin against his servant David, because he has not sinned against you, and because his deeds have brought good to you. 5 For he took his life in his hand [וַיָּשֶׂם֩ אֶת־נַפְשֹׁ֨ו בְכַפֹּ֜ו וַיַּ֣ךְ MT, καὶ ἔθετο τὴν ψυχὴν αὐτοῦ ἐν τῇ χειρὶ αὐτοῦ, LXX] and he struck down the Philistine, and the LORD worked a great salvation for all Israel. You saw it, and rejoiced. Why then will you sin against innocent blood by killing David without cause?” (1 Sam. 19:4–5, ESV)
Note how David compared this to shepherding:
And Saul said to David, “You are not able to go against this Philistine to fight with him, for you are but a youth, and he has been a man of war from his youth.” 34 But David said to Saul, “Your servant used to keep sheep for his father. And when there came a lion, or a bear, and took a lamb from the flock, 35 I went after him and struck him and delivered it out of his mouth. And if he arose against me, I caught him by his beard and struck him and killed him. 36 Your servant has struck down both lions and bears, and this uncircumcised Philistine shall be like one of them, for he has defied the armies of the living God.” 37 And David said, “The LORD who delivered me from the paw of the lion and from the paw of the bear will deliver me from the hand of this Philistine.” (1 Sam. 17:33–37, ESV)
Perhaps Jesus left out “into his hands” to leave it with the ambiguity to refer to sacrificing his life, but his listeners would see a good shepherd as risking his life.
Another interesting possibility is when one gets Google translate to translate “he lays down his life” in Hebrew הוא מקדיש את חייו. The participle מקדיש means devote, and the root means קדש means holy. Thus, it could mean the devotion of the shepherd and also refer the sin offering. However, I have no idea why Google did this translation or how if any it connects to a first century meaning.