If the keys are all independent and near-uniform random, then an chosen-plaintext attacker has near-zero advantage in the IND-CPA game.
In the IND-CPA game, the attacker can ask for the ciphertexts of any plaintexts of their choice (CPA, Chosen-Plaintext Attack), and then submits two challenge messages, and, given the ciphertext for one of challenge messages, wins the game if they can tell which one (IND for indistinguishability). The attacker's advantage is their probability above 1/2 of winning this game. This is the standard notion of security for an unauthenticated cipher.
Of course, this only thwarts an eavesdropper who can listen but not touch the messages. A MITM could easily transform one ciphertext into another, forging messages without raising any alarms, even if they can't learn anything about the plaintext given the ciphertext—they may know a priori what the plaintext will be and may want to replace it by something else.
So this is usually not enough security for applications that involve, say, the internet: you need an authenticated cipher for that, for example a one-time pad with a one-time authenticator appended, which is the model that AES-GCM and NaCl crypto_secretbox_xsalsa20poly1305 make practical with short keys expanded pseudorandomly into message-length pads.