UART: two transmitters, single slave

You can definitely transmit data using just TX & GND.

Firstly, you want to hook up the ATtiny85 TX line to the FTDI RX line (yellow on the TTL-232R). Make sure that the USB adapter can handle 5V - I'm fairly sure even the 3.3V TTL-232R is 5V tolerant.

According to the example page for SoftwareSerial, you need to set the direction of the TX & RX lines in your setup function:

// include the SoftwareSerial library so you can use its functions:
#include <SoftwareSerial.h>

#define rxPin 2
#define txPin 3
#define ledPin 13

// set up a new serial port
SoftwareSerial mySerial =  SoftwareSerial(rxPin, txPin);
byte pinState = 0;

void setup()  {
  // define pin modes for tx, rx, led pins:
  pinMode(rxPin, INPUT);
  pinMode(txPin, OUTPUT);
  pinMode(ledPin, OUTPUT);
  // set the data rate for the SoftwareSerial port
  mySerial.begin(9600);
}

The baudrate will be 4800 in your case. The SoftwareSerial library doesn't seem to support CTS & RTS, so just make sure you aren't using them on the host software.

Check out the reference page for more details, where they talk about some potential timing issues which may be exacerbated if you're running at 1MHz using the internal oscillator on the tiny.

I dont understand your confusion. You want to communicate between the two controllers with UART, then ofcourse both of them need UART capabilities and code for this. Your computer has drivers and software for dealing with this, so your microcontroller also needs a driver for it.

Your ATmega32 need UART TX functionality, while AT90USB1287 need UART RX functionality. Having already written both TX and RX for your ATmega32, writing the UART RX routine for AT90 should be a breese. You could add a bitvalue of some kind, identifying that the UART RX on AT90 is coming from your ATmega32, and directly pass it through to USB.

EDIT: Code from AVRFreaks that is written for at90usb1287

#include <avr/io.h> 
#include <util/delay.h> 

#define RXC      (RXC1) 
#define TXC      (RXC1) 
#define UCSRC    (UCSR1C) 
#define UCSRB    (UCSR1B) 
#define UCSRA    (UCSR1A) 
#define UDR      (UDR1) 
#define UBRRL    (UBRR1L) 
#define UBRRH    (UBRR1H) 
#define UBRR     (UBRR1) 
#define UDRE     (UDRE1) 
#define RXCIE    (RXCIE1) 
#define TXEN     (TXEN1) 
#define RXEN     (RXEN1) 
#define UCSZ0    (UCSZ10) 
#define UCSZ1    (UCSZ11) 
#define UMSEL0   (UMSEL10) 
#define UMSEL1   (UMSEL11) 

#define FRAMING_ERROR (1<<FE1) 
#define PARITY_ERROR (1<<UPE1) 
#define DATA_OVERRUN (1<<DOR1) 
#define DATA_REGISTER_EMPTY (1<<UDRE1) 
#define RX_COMPLETE (1<<RXC1) 

#define FOSC 16000000 

typedef unsigned char       Uchar; 
typedef unsigned long int   Uint32; 

#define Wait_USART_Ready() while (!(UCSR1A & (1<<UDRE1))) 

#define lowByte(w) ((uint8_t) ((w) & 0xff)) 
#define highByte(w) ((uint8_t) ((w) >> 8)) 

int main(void) 
{ 
/* Disable clock division */ 
clock_prescale_set(clock_div_1); 
   UBRRH = (Uchar)((((Uint32)FOSC)/((Uint32)USART_BAUDRATE*16)-1)>>8); 
   UBRRL = (Uchar)(((Uint32)FOSC)/((Uint32)USART_BAUDRATE*16)-1) & 0x0ff; 

   UCSRB |= (1 << RXEN) | (1 << TXEN); 
   UCSRC |= (1 << UCSZ1) | (1 << UCSZ0); 

   char ReceivedByte; 
   for (;;) // Loop forever 
   { 
      while ((UCSRA & (1 << RXC)) == 0) {}; // Do nothing until data have been received and is ready to be read from UDR 
      ReceivedByte = UDR; // Fetch the received byte value into the variable "ByteReceived" 

      while ((UCSRA & (1 << UDRE)) == 0) {}; // Do nothing until UDR is ready for more data to be written to it 
      UDR = ReceivedByte; // Echo back the received byte back to the computer 
   } 
}