Efficiency is of course a relative term. Usually lattice-based systems are considered bandwidth inefficient in comparison to, say, RSA and elliptic curve systems where cryptograms are typically 1000s and 100s of bits respectively. For example for a notional level of security where breaking the cryptograms takes as much work as breaking AES-128 with a classical computer (128-bits of exhaustion work), it is widely held that an RSA modulus of 3072-bits or an elliptic curve group of 256-bits would be necessary.
There's some debate as to the correct way to assess the cost of attacks on lattice systems, but one can talk in broad brush strokes about where the increased bandwidth requirement comes from. Essentially, almost all lattice based cryptography will have a cryptogram where of the form an $n$-long vector of integers of size less than some value $q$. Such a cryptogram will be at least $n\lg q$-bits in size. The security of the system will rely on the hardness of finding a short vector among a lattice generated by similarly sized vectors. In an asymptotic and not very reliable sense, many people believe that it is possible to find such a vector with roughly $0.292n$-bits of work (see The General Sieve Kernel and New Records in Lattice Reduction by Albrecht et al for a deep technical survey). This coarse estimate suggests that $n$ should be taken at least 438 for the AES-128 level of security (and if we look at proposal such as Kyber, NTRU and SABER that aim to hit this level of security they typically have $n=500-512$).
The choice of $q$ is also somewhat arcane. Early proposals suggested taking $n^2<q<2n^2$, though recent parameter sets are closer to $q\approx n\log n$. I can't recall ever seeing a proposal with $q$ less than $n$ (ETA: Mark points out that the NIST 2nd round candidate and Chinese CACR choice algorithm LAC does have $q<n$). For the systems mentioned above $q$ is between 11 and 13 bits. The bandwidth for a single vector is now looking like 7144-bits of any scheme that aspires to the same level of (classical) security as AES-128 (these numbers are all very ballpark and should not be mistaken for a rigorous analysis). This is more than twice the bandwidth of RSA which many already consider quite heavyweight in terms of bandwidth. It is perhaps worth noting that for higher levels of security, the bandwidth requirements of RSA grow faster than those of lattice systems, but both look very inefficient in comparison to elliptic curves.
This analysis applies only to cryptograms and there are also questions as to the costs of transmitting public keys or systems where more information than a single vector is required. It should give some sense of the issues however.
