ChaCha20 and Salsa20 are stream ciphers. Their purpose is to provide confidentiality, as long as the key used to seed them remains secret. Poly1305 however is a Message Authentication Code (MAC), which is supposed to provide integrity and authenticity for the sent messages.
In general, it should be possible to use them separately as they do not seem to depend on each other (Poly1305 can, and does as far as TLS is concerned, use ChaCha20 as an underlying cipher, however). For example, the NaCl crypto library lets you do encryption using Salsa20, authentication using Poly1305 or authenticated encryption using both schemes.
However, it is not possible to use them separately with TLS. The reason for this is, that TLS uses a scheme to identify the combinations of Key Exchange Algorithm/Cipher/MAC called "Cipher Suites". As a client always sends all the cipher suites it supports in each "Client Hello" packet (the first packet needed to establish a TLS connection), all possible cipher suites are predefined, so a 2-byte ID is sufficient to represent a single cipher suite.
As RFC7539 states, the main reasons for including the possibility to use ChaCha20 and Poly1305 are to have an alternative to AES in case weaknesses are found and to have a fast alternative to use on slow hardware [Page 2].
You can see all the defined cipher suites and their representations at IANA's TLS parameter specification. Alternatively, have a look at RFC7539 (link at schroeder's comment to your post), which specifies all possible cipher suites using ChaCha/Poly.