Why did the Eastern Gate of Moria not have password protection, like the Western Gate with its “speak friend ‘riddle’”? Were the peoples on the east side considered more trustworthy?
The Eastern gate had other protections
According to notes on by Tolkien, in an untitled essay, described by Tolkien as:
An extensive commentary and history of the interrelation of the languages in The Silmarillion and The Lord of the Rings, arising from consideration of the Book of Mazarbul, but attempting to clarify and where necessary to correct or explain the references to such matters scattered in The Lord of the Rings, especially in Appendix F and in Faramir's talk in LR II.
Tolkien discusses (in a note on the essay about runes and their origin) the protections that lay on the West gate. Tolkien describes spells preventing those without the leave of the Lord from entering.
But the East Gates, which perished in the war against the Orks, had opened upon the wide world, and were less friendly. They had borne Runic inscriptions in several tongues: spells of prohibition and exclusion in Khuzdul, and commands that all should depart who had not the leave of the Lord of Moria written in Quenya, Sindarin, the Common Speech, the languages of Rohan and of Dale and Dunland.
The History of Middle-earth - Volume XII, The Peoples of Middle-earth: "Of Dwarves and Men", note 8
The East gate was the original gate, built when Durin the Deathless and first discovered Moria after looking into the Mirrormere early in the First Age. It was the only way in and out and would've been heavily guarded like the entrance to any Kingdom/Fortress.
The West gate was only built in the Second Age, by Narvi the Dwarf and Celebrimbor the Elf, for use by the Noldorin Elves of Eregion in their trade with the Dwarves. It was unguarded and really simply an opening into the tunnel, left open until Sauron took control of Eregion, when the gate was shut.
In addition to the spells on the actual gate mentioned in Edlothiad's answer, there was a secondary line of defence behind the gate (bold added):
Suddenly Frodo saw before him a black chasm. At the end of the hall the floor vanished and fell to an unknown depth. The outer door could only be reached by a slender bridge of stone, without kerb or rail, that spanned the chasm with one curving spring of fifty feet. It was an ancient defence of the Dwarves against any enemy that might capture the First Hall and the outer passages. They could only pass across it in single file. At the brink Gandalf halted and the others came up in a pack behind.
The Lord of the Rings (The Fellowship of the Ring, Book 2). Chapter 5: The Bridge of Khazad-Dûm
Any invading force that managed to get past the gate would be reduced to single file and easily picked off by the defenders. As a last resort the bridge could be smashed. (As Gandalf eventually does.) Anyone trying to sneak in would have to cross the bridge where they would be exposed (assuming they're not invisible) and easily spotted by any guard.