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In TLS separate implicit sequence numbers for sent and received packets are used to calculate the HMAC in the record layer. The RFC 5246 says:

Each connection state contains a sequence number, which is maintained separately for read and write states.

Why do I need separate sequence numbers? Why can't two parties share the same one?

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TLS is for bidirectional traffic, i.e. there are separate data streams for each direction which can have a different position in the data stream. That's why one also needs two sequence numbers to describe these positions. Same as with TCP.

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Because the two sides don't take turns transmitting. Both sides transmit at the same time.

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