The answer of usr is still correct in 2020, but for those wondering about why it is not trivial here is a short explanation.
The idea behind the steams is that writing to the stream and reading from it are independent. Usually, the process of reading is much faster then writing (think about receiving data through network - you can read the data as soon as it arrives) so usually the reader waits for new portion of data, processes it as soon as it arrives and then drops it to free the memory, and then waits for the next portion.
This allows processing potentially infinite data stream (for example, application log stream) without using much RAM.
Suppose now we have 2 readers (as required by the question). A data portion arrives, and then we have to wait for both the readers to read the data before we can drop it. Which means that it must be stored in memory until both readers are done with it. The problem is that the readers can process the data with very different speed. E.g. one can write it to a file, another can just count the symbols in memory. In this case either the fast one would have to wait for the slow one before reading further, or we would need to save the data to a buffer in memory, and let the readers read from it. In the worst case we will end up with the full copy of the input stream in memory, basically creating an instance of memory stream.
To implement the first option, you would have to implement a stream reader that is aware which of your stream usage is faster, and considering it, would distribute and drop the data accordingly.
If you are sure you have enough memory, and processing speed is not critical, just use the memory stream:
using var memStream = new MemoryStream();
await incomingStream.CopyToAsync(memStream);
UseTheStreamForTheFirstTime(memStream);
memStream.Seek(0, SeekOrigin.Begin);
UseTheStreamAnotherTime(memStream);