If I have a int32 type integer in the 8-bit processor's memory, say, 8051, how could I identify the endianess of that integer? Is it compiler specific? I think this is important when sending multybyte data through serial lines etc.
With an 8 bit microcontroller that has no native support for wider integers, the endianness of integers stored in memory is indeed up to the compiler writer.
The SDCC compiler, which is widely used on 8051, stores integers in little-endian format (the user guide for that compiler claims that it is more efficient on that architecture, due to the presence of an instruction for incrementing a data pointer but not one for decrementing).