The protocol as in the question, and [SRA1979]/[SRA1981], are insecure as described, but smalls modifications can make it secure.
A serious security issue is that the encryption used is such that for any message $m$ (e.g. the description of a card) and any prime factor $p$ of $n$ (i.e. $p=n$ when $n$ is prime), ciphertext $c=m^e\bmod n$ has the property that the Legendre symbols $\left(\frac m p\right)$ and $\left(\frac c p\right)$ are equal (either $-1$ or $+1$). That is, $c^{(p-1)/2}\bmod p=m^{(p-1)/2}\bmod p$. This allows some level of cheating. For prime $n$, that's much like for a physical deck of cards where the orientation of a card is recognizable by it's back, and the orientation of each individual card is about random, but known. Composite $n$ only makes things worse, since the factors of $n$ are public and the attack can be carried for each.
There is another lesser security issue, related to the Desmedt and Odlyzko attack [DO1985]: the factorization of the $m_i$ representing cards conceivably could be such that we can find some not-all zero tuple of $t_i\in\mathbb Z$ such that
$$\prod_{t_i>0}{m_i}^{t_i}\,=\,\prod_{t_i<0}{m_i}^{-t_i}$$
and that would allow to identify the cards with non-zero $t_i$ in this tuple from their ciphertexts $c_i$ (often exactly, perhaps within a few permutations). The probability is low for non-malicious choice of $m_i$, but the ability to subtly alter the $m_i$ (e.g. by inserting invisible characters at the end…) would make it totally practical.
Update: I think [L1979] and [L1981] note the first attack. [C1985] extends it for $n$ with $n-1$ divisible by small odd primes, and gives a variation of the second attack.
I assert without proof that we can make the protocol secure with the following small modifications starting from [SRA1981]:
Choose $n$ a haphazard safe prime of 2048-bit or more, e.g. this or this. That also simplifies the choice of the encryption exponent $e$: we only need $e$ to be odd, large enough ($\lfloor\log_2 n\rfloor$ bit is fine, even 300 bits is enough), otherwise about uniformly random, and secret. We can compute the decryption exponent $d$ as $e^{-1}\bmod(n-1)$, that is also $e^{n-4}\bmod(n-1)$ or $e^{(n-5)/2}\bmod(n-1)$.
Obtain $m_i$ representing card $i$ as $m_i=H(\underline i)^2\bmod n$, where
- $\underline i$ represents $i$ as byte(s) per some fixed convention (e.g. the byte with value $i$, or the ASCII representation of the text suggested by [SRA1981] for card $i$)
- $H$ is a cryptographic hash/PRF function with output of size $\lfloor\log_2 n/8\rfloor$ bytes (e.g. $H(M):=\operatorname{SHAKE256}(M,8\lfloor\log_2 (n/8)\rfloor)$ as defined in FIPS 202)
- we assimilate the bitstring output of $H$ to integer per some convention (e.g. big-endian).
Decryption is slightly modified: we do not obtain the decryption of the card itself, but some $m_i$. Since these are public, we can search which card it is.
Rationale: Precaution 1 makes the Discrete Logarithm Problem modulo $n$ hard, and is recommended in [C1985] to guard against attacks exploiting small factors of $p-1$. That works because precaution 2 is such that the order of $m_i$ is the prime $(p-1)/2$. Further, the hash guards against any variant of the Desmedt and Odlyzko attack, much like in Full Domain Hash [C2000].
Extension: If the number of "cards" is large to the point of making search inconvenient, or there was no agreed-upon description of cards, we can use $m_i=\operatorname{pad}(M_i)$ where $M_i$ is the description of card, and $\operatorname{pad}$ is a reversible pseudo-random encoding like that of ISO/IEC 9796-2 scheme 3 in total recovery mode.
There are many later mental poker protocols requiring less computational work and bandwidth. See [WW2009] for a bibliography, and another one.
[SRA1979]: Adi Shamir, Ronald Rivest, Leonard Adleman; Mental Poker, MIT-LCS-TM-125, January 1979.
[L1979] R. Lipton, How to Cheat at Mental Poker, Technical Report, Computer Science Depertment, Berkeley, CVA, August 1979 (no online source found).
[L1981] R. Lipton, How to Cheat at Mental Poker, in Proceeding of the AMS short course on Cryptology, 1981 (no online source found).
[SRA1981]: Adi Shamir, Ronald Rivest, Leonard Adleman; Mental Poker, in The Mathematical Gardner, 1981.
[C1985]: Don Coppersmith; Cheating at Mental Poker, in proceedings of Crypto 1985 (note: read $a\,\bar a\equiv1\pmod{n-1}$ where there is $a\,\bar a\equiv1\pmod n$).
[DO1985]: Yvo Desmedt and Andrew M. Odlyzko; A chosen text attack on the RSA cryptosystem and some discrete logarithm schemes, in proceedings of Crypto 1985.
[C2000]: Jean-Sébastien Coron; On the Exact Security of Full Domain Hash, in proceedings of Crypto 2000.
[WW2009]: Tzer-jen Wei, Lih-Chung Wang; A Fast Mental Poker Protocol, eprint 2009/439.
