If our left Fallopian tube was removed then we...

Dear Customer
Welcome, and thank you for trusting us
Regarding your question
advice:
no it does not need to lie on the right side 
how pregnancy occurs:

• at time of ovulation, ovum is released form the ovary 
• then it is picked up by fallopian tube in the same side
• after that it is transmitted into the tube till it find the sperm, fertilization occurs.
• then zygot transported to uterus and implanation occurs.

can right fallopian tube take ovum from left ovary and vice versa?
no it picked ovum in the same side.
in your case:

• you removed left tube, so if ovulation occurs in left side ovum will not be picked 
• if it occurs in right ovary it will be oicked by right tube .

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