It would definitely be possible. It would not, however, have its own orbit.
Right now, the inner solar system is pretty full, gravitationally speaking, without much in terms of room between the inner planets for more additional planets. Your new planet, Terra II, would have to share its orbit with one of the existing planets.
How can two planets share an orbit?
In our solar system, most things that share an orbit with other things do so either as moons of other planets, or by hanging out in their Lagrange points, like the Trojan asteroids do with Jupiter. You could stick an Earth-sized planet in the L4 or L5 Lagrange points, and it would be gravitationally stable.
This isn't however, the most likely way for Terra II to come to exist. While stable, the Lagrange points don't tend to be home to large bodies, probably due to the fact that the formation of a large body involves lots of energetic collisions, which are great for knocking things out of subtle gravity wells like the Lagrange points.
What's more likely is that Terra II would act as a binary partner for one of the other inner planets, such as Venus or Earth. Picture the Moon, if you will. Now picture it the size of Earth. Both planets would be large enough for an atmosphere, and the binary system would be gravitationally stable. While I don't know of any planets that have been found (so far) in such a system, we've found stars that orbit each other like this. The Castor system, for example, consists of two pairs of binary stars that orbit each other, and which are in turn all orbited by a more distant pair of binary stars. Planets could do the same.
How would we communicate?
Well, if both planets had intelligent life at the same time, we could do so with radio waves. We'd probably have sent a manned mission there, as well, since "We're talking to some guys on Mars and want to go say hi," is a more compelling argument than, "We want to make human footprints there, because America."
Unfortunately, intelligent life is extremely rare. It's more likely that only one planet would be inhabited by sentient creatures. We'd still have some strongly compelling reasons to go visit, but it's unlikely that we'd find anything there that resembles us.