No, the caster does not magically know if their message was received.
Since spells only do what they say, and Sending does not say that the caster knows if the message was properly received, the caster does not know. The text also alludes to it, saying "... there is a 5% chance that the message doesn't arrive." Not that it didn't send, or that the spell failed. You successfully sent the message, it just got lost. This is true whether or not you're on a different plane - you get no verification that it was received when you're on the same plane either, you just know that if it sent, it was received on this plane (Thanks, @MivaScott).
Obviously, since it doesn't say it explicitly, the DM can rule differently.
Also, keep in mind that the creature can respond. If the caster is expecting a response and doesn't receive one, they can at least assume something went wrong (although that doesn't necessarily mean the 5% chance happened, the creature could be dead or just didn't want to respond).