As pointed in [comment][1]

1. _"encrypt" with sender's private key_ is improper terminology; the proper term is _sign_.
2. it's assumed $n_r>n_s$ so that $c_1<n_r$, which is necessary to get $\widetilde{c_1}=c_1$. Thus the technique won't "work" with 100% reliability in both directions.

Also, the issue of padding is ignored. And more broadly the goal is not stated.

Issue 2 can be solved in a number of ways
- The best is to use separate keys for signature and encryption, with a maximum $n$ for the first applied (signature in the question) lower than the minimum for the second. That also abides to the time-proven principle: _one usage, one key_. It also avoids ending up with $c_2=m$ when the sender is the receiver, something an adversary might induce.
- Another option is to force all $n$ to share the same bit size and many high order bits (say 512, which become a fixed public arbitrary constant). That insures $c_1<n_r$ with overwhelming probability. It does not solve the send-to-self issue.
- Yet another is to require all $n$ to have the same bit size, restrict to (say) even $m$, and use $c_1\gets \min\bigl((m^{d_s}\bmod n_s),(-m^{d_s}\bmod n_s)\bigr)$. On decryption, we'll have either $m=\widetilde{c_2}$ or $m=n-\widetilde{c_2}$, and the parity of $m$ allows to determine which. It does not solve the send-to-self issue.
- A variant of the above forces $m<\min(n)/2$, which also allows recovery of $m$.

Another issue is that secure RSA encryption (for the proper meaning of that, not signing, where that's optional) requires adding randomness when encrypting. And that both RSA encryption and signing require redundancy for security. For this reason, it's best to proceed in the order:
- pad $m$ with randomness and redundancy
- encrypt with textbook RSA using public key $(n_r,e_r)$ [in it's instance for encryption, if distinct from signature]
- sign with textbook RSA using private key $(n_s,e_s)$ [in it's instance for signature]

An example of that is in the [RSA-based authentication protocol][2] between Smart Card and (pan-)European Digital Tachograph (designed circa the end of the 20<sup>th</sup> century, I wish I knew by who). It uses the $\min$ and even padded message trick. A modern variant would use [RSAES-OAEP][3] padding and the $m<\min(n)/2$ trick, or best separate encryption and decryption keys.


  [1]: https://crypto.stackexchange.com/posts/comments/199064
  [2]: https://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:32016R0799&from=EN#page=351
  [3]: https://pkcs1.grieu.fr/#page=16