Short answer: Yes, clients and servers can send requests and responses at the same time.
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→
GET /request1.html HTTP/1.1
Host: example.com
...
GET /request2.html HTTP/1.1
Host: example.com
...
GET /request3.html HTTP/1.1
Host: example.com
...
→
HTTP/1.1 200 OK
Content-Length: 234
...
HTTP/1.1 200 OK
Content-Length: 123
...
HTTP/1.1 200 OK
Content-Length: 345
...
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